Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES BRANDON K OPR O WSKI AND JOEL BREWSTER LEWIS A B S T R AC T . W e consider the problem of enumerating hypermatrices of format 2 × ( k + 1) × k over a ﬁnite ﬁeld that hav e nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. W e conjecture an attracti v e product formula for the enumeration, and prov e it in many cases. In general, we show that the enumeration is gi ven (up to a po wer of q − 1 ) by a polynomial in q with nonnegativ e integer coef ﬁcients, whose value at q = 1 enumerates a natural family of three- dimensional rook placements. 1. I N T RO D U C T I O N In this paper we attempt to generalize the classical q -rook theory of matrices to higher dimensions. W e be gin with a brief surve y of the classical theory which will serve as a starting point for our generalization. Counting permutations as rook placements was ﬁrst systematically studied by Kaplansky and Ri- ordan [KR46], who showed a large family of permutation enumeration questions can be phrased in the terms of rook theory . Here a rook can be thought of as a chess rook (which can attack verti- cally and horizontally) and a rook placement is a placement of non-attacking rooks on a subset of a chessboard; for a more precise statement see Deﬁnition 2.7. Rook theory was further dev eloped in a series of ﬁv e papers by Goldman, Joichi, and White. In particular , their ﬁrst paper [GJW75] de veloped a complete understanding of when two partition-shaped boards have the same number of rook placements for all numbers of rooks. The “ q ” in q -rook theory was ﬁrst studied by Garsia and Remmel [GR86], who introduced the q -rook number R k ( B ; q ) = X C q inv( C,B ) of a board B . In this formula, the sum is over the set of rook placements C of k rooks on the partition-shaped board B , and in v( C, B ) is a certain in version statistic associated to each placement. In addition to rook numbers, one can also look at hit numbers for a particular board contained in the n -by- n square: the k th hit number is the number of n -rook placements on the square in which exactly k rooks are placed in B . Garsia and Remmel were using the language of rook theory as a means to solve a separate problem but in the process discovered that the q -hit number for partition- shaped boards is a polynomi al with positi v e coef ﬁcients. Howe ver , the y were unhapp y with the proof of this fact and asked for others to pro vide a more satisfying e xplanation. In response to this call for a better proof, Haglund [Hag98] gav e an interpretation of q -rook and q -hit numbers in terms of matrices over ﬁnite ﬁelds with entries forced to be zero. 1 In particular, Haglund devised a way to relate each placement of k rooks on a board B to a collection of matrices ov er a ﬁnite ﬁeld F q of rank k . He then used this to prove that the number of n × n matrices of rank k with support corresponding to B , which he called P k ( B ) , is related to the q -rook number R k ( B ; q ) by (1) P k ( B ) = ( q − 1) k q | B |− k R k ( B ; q − 1 ) . In the particular case that k = n , the expression on both sides of this equation factors in an attracti v e way (see Theorem 2.10). Date : February 25, 2026. 1 An alternativ e proof, via other methods, was gi v en around the same time by Dworkin [Dwo98]. 1 2 B. KOPR OWSKI AND J. B. LEWIS The main goal of this paper is to generalize this connection between the number of rook placements and the number rank- k matrices ov er a ﬁnite ﬁeld to higher dimensions by enumerating hypermatri- ces . Our work is inspired by an unpublished note by Musiker and Y u [MY08], who showed that the number of hypermatrices (see Deﬁnition 2.1) ov er F q with nonzero hyper determinant , the multidi- mensional analogue of the determinant, factors nicely in the 2 × 2 × 2 case. It is worth noting that there are at least three inequi v alent ways to deﬁne a hyperdeterminant; we will be using Cayley’ s second deﬁnition, commonly denoted Det , whose important properties are outlined below in Corollary 2.4. If a hypermatrix has nonzero hyperdeterminant we say it is nondegenerate . W e denote by GL k ( F q ) the general linear group of k × k in vertible matrices over F q . It is easy to show that | GL 2 ( F q ) | = ( q 2 − 1)( q 2 − q ) . If we let [ n ] q = 1 + q + q 2 + · · · + q n − 1 and [ n ]! q = [ n ] q [ n − 1] q · · · [1] q then we can write this more succinctly as | GL 2 ( F q ) | = q ( q − 1) 2 [2]! q . Musiker and Y u prov ed the number of nondegenerate 2 × 2 × 2 hypermatrices o ver F q is ( q 4 − 1)( q 4 − q 3 ) , which led them to conjecture the number of nondegenerate hypermatrices of dimension k + 1 in the general 2 × 2 × · · · × 2 case is ( q 2 k − 1)( q 2 k − q 2 k − 1 ) . Howe v er , they were unsuccessful in e v en proving the 2 × 2 × 2 × 2 case due to the fact that the hyperdeterminant is this case is unapproachable, having o v er 2 million terms. It turns out that, rather than the “cubic” hypermatrices considered by Musiker and Y u, the easier hypermatrices to study are those of boundary f ormat , which are hypermatrices of shape ( k 1 + 1) × ( k 2 + 1) × · · · × ( k r + 1) , with k i = k 1 + k 2 + · · · + k i − 1 + k i +1 + · · · + k r for some 1 ≤ i ≤ r . It was shown by Aitken [Ait19] that for dimension 3 , the question of whether or not a boundary format hypermatrix is nonde generate reduces to a question of whether or not a certain family of matrices are full rank (see Lemma 2.5). This is helpful for enumeration but it is actually better than that: using this fact Aitken showed that there exists a free and transitiv e group action on the set M k ( F q ) of nondegenerate 2 × ( k + 1) × k hypermatrices with entries in F q and using this he was able to sho w that | M k ( F q ) | = q k 2 ( q − 1) 2 k [ k + 1]! q [ k ]! q . W e pick up where Aitken left off, reﬁning this count for boundary format hypermatrices with re- stricted entries. W e be gin in Section 2 by re vie wing the basic deﬁnitions and concepts of hypermatrices and the hy- perdeterminant. W e then sho w there exists an analogous concept to matrices with an integer partition- shaped section forced to be zero for boundary format hypermatrices. Speciﬁcally , these are h yperma- trices with a plane partition section forced to be zero. W e also show that there is a unique maximal (by inclusion) plane partition such that nondegenerate hypermatrices e xist when the entries of the plane partition are forced to be zero. In Section 3, we state our main conjecture (Conjecture 3.1): for 2 × ( k + 1) × k hypermatrices and a plane partition P = ⟨ λ, µ ⟩ where λ and µ are integer partitions of length k (see Deﬁnition 2.11 for this notational con v ention), the number of nondegenerate h ypermatrices with support av oiding P is q k 2 ( q − 1) 2 k [ k + 1 − λ 1 ] q · · · [2 − λ k ] q · [ k − µ 1 ] q · · · [1 − µ k ] q . Although we are unable to prov e our conjecture for all P , we sho w that it holds for se v eral natural families of plane partitions, including plane partitions of one layer , the largest and smallest plane partitions for which the question is interesting, and all plane partitions in the case k = 2 . In Section 4, we provide further indirect evidence in fa v or of Conjecture 3.1. W e introduce a three-dimensional analogue of rook placements that we call hyperr ook placements . W e show that the ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 3 number of hyperrook placements on a 2 × ( k + 1) × k board av oiding a plane partition P = ⟨ λ, µ ⟩ is exactly the v alue ( k + 1 − λ 1 ) · · · (2 − λ k ) · ( k − µ 1 ) · · · (1 − µ k ) that one w ould predict by substituting q = 1 into Conjecture 3.1. Further , in Section 4.2, we make a concrete connection between these hyperrook placements and nondegenerate hypermatrices con- crete, giving an analogue of Haglund’ s theorem (1). Our main result (Theorem 4.1) is to establish the following weak version of Conjecture 3.1: for ev ery plane partition P , the number of nondegen- erate hypermatrices over F q av oiding P is giv en by a polynomial in q that (up to factors of q − 1 ) has positi ve-inte ger coef ﬁcients and whose v alue at q = 1 is the number of hyperrook placements av oiding P . W e conclude the paper with Section 5 with some ﬁnal remarks and open questions. 2. H Y P E R M A T R I C E S A N D N O N D E G E N E R AC Y Deﬁnition 2.1. A hypermatrix of format ( k 1 + 1) × ( k 2 + 1) × · · · × ( k r + 1) ov er a ﬁeld F is an r -dimensional array of elements a i 1 ,...,i r ∈ F with 1 ≤ i j ≤ k j + 1 for j = 1 , . . . , r . A slice of a hypermatrix is an ( r − 1) -dimensional subarray which is obtained by holding one index constant. Furthermore, a slice in the i th direction is the subarray where we hold the i th index constant. F or instance, the 2 × ( k + 1) × k hypermatrix ( a i 1 ,i 2 ,i 3 ) with 1 ≤ i 1 ≤ 2 , 1 ≤ i 2 ≤ k + 1 , and 1 ≤ i 3 ≤ k has two ( k + 1) × k slices in the ﬁrst direction: ( a 1 ,i 2 ,i 3 ) and ( a 2 ,i 2 ,i 3 ) . For a hypermatrix of dimension ( k 1 + 1) × ( k 2 + 1) × · · · × ( k r + 1) we can see there are k i + 1 slices in the i th direction, each of dimension ( k 1 + 1) × · · · × ( k i − 1 + 1) × ( k i +1 + 1) × · · · × ( k r + 1) . W e call any two distinct slices in the same direction parallel . In this paper we will be especially concerned with hypermatrices of boundary format 2 × ( k + 1) × k which we will vie w as the pair ( M 1 , M 2 ) of ( k + 1) × k slices in the 1 st direction. W e will sometimes refer to M 1 as the fr ont face and M 2 as the back face . F or instance, the follo wing is a representation of a 2 × 4 × 3 hypermatrix:     1 0 0 0 1 0 0 0 1 0 0 0         0 0 0 1 0 0 0 1 0 0 0 1     . The collection of ( k 1 + 1) × ( k 2 + 1) × · · · × ( k r + 1) hypermatrices o ver a ﬁeld F comes equipped with the natural action of the group G = GL k 1 +1 ( F ) × GL k 2 +1 ( F ) × · · · × GL k r +1 ( F ) , deﬁned as follo ws. Let H be a hypermatrix of format ( k 1 + 1) × · · · × ( k r + 1) with A i ∈ GL k i +1 ( F ) and M 1 , M 2 , . . . , M k i +1 the slices of H in the i th direction. Let A = ( a j 1 ,j 2 ) with 1 ≤ j 1 , j 2 ≤ k i +1 , then (1 , . . . , A i , . . . 1) ◦ H is the hypermatrix with slices M ′ 1 , M ′ 2 , . . . , M ′ k i +1 in the i th direction where M ′ n = a n, 1 M 1 + a n, 2 M 2 + · · · + a n,k i +1 M k i +1 . In other words, (1 , . . . , 1 , A i , 1 , . . . , 1) ∈ G acts on H by taking linear combinations of the k i + 1 slices in the i th direction in the coefﬁcients of A i . In the case of 2 × ( k + 1) × k hypermatrices represented as a pair of ( k + 1) × k matrices, the GL k +1 ( F q ) -action looks like simultaneous ro w 4 B. KOPR OWSKI AND J. B. LEWIS operations on M 1 and M 2 , and the GL k ( F q ) -action looks like simultaneous column operations. For instance,      1 0 0 1  ,     a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 d     ,   0 0 x 1 0 0 0 1 0       ◦     1 0 0 0 1 0 0 0 1 0 0 0         0 0 0 1 0 0 0 1 0 0 0 1     =     0 a 0 0 0 b cx 0 0 0 0 0         0 0 0 0 b 0 0 0 c dx 0 0     . There is a function on hypermatrices called the hyperdeterminant that plays a similar role to the determinant for matrices. In particular we call hypermatrices with zero hyperdeterminant degener - ate and correspondingly hypermatrices with nonzero hyperdeterminant nonde generate. T o distiguish between the hyperdeterminant and matrix determinant we will denote the hyperdeterminant of a hy- permatrix M by Det( M ) and determinant of a matrix A by det( A ) . The formal deﬁnition of the hyperdeterminant is technical and outside the scope of this paper so instead we will describe its prop- erties; for a complete deﬁnition see [GKZ94, p. 444]. Although the hyperdeterminant is deﬁned for hypermatrices of an y format, there are numerous formats for which e very hypermatrix is degenerate, as the next theorem sho ws. Theorem 2.2 ([GKZ94, Thm. 1.3]) . The hypdeterminant of format ( k 1 + 1) × · · · × ( k r + 1) is non-trivial if and only if k j ≤ X i  = j k i for all j = 1 , . . . , r . W e no w restrict our attention to h ypermatrices of non-tri vial format. Let us ﬁx a non-tri vial format ( k 1 + 1) × ( k 2 + 1) × · · · × ( k r + 1) . Proposition 2.3 ([GKZ94, Prop. 1.4]) . The hyper determinant is relatively in variant under the gr oup action G = GL k 1 +1 ( F ) × · · · × GL k r +1 ( F ) , that is, ther e e xist inte gers n 1 , n 2 , . . . , n r such that if M is a hypermatrix and ( A 1 , . . . , A r ) ∈ G then Det  ( A 1 , . . . , A r ) ◦ M  = det( A 1 ) n 1 · · · det( A r ) n r · Det( M ) . The pre vious proposition leads to the follo wing corollaries; part (d) in particular will be helpful when we classify hypermatrices with certain entries forced to be zero. Corollary 2.4 ([GKZ94, Cor . 1.5]) . (a) Inter c hanging two parallel slices leaves the hyper determinant in variant up to sign. (b) The hyperdeterminant is a homogeneous polynomial in the entries of each slice. The de gr ee of homogeneity is the same for par allel slices. (c) The hyper determinant does not c hange if we add to some slice a scalar multiple of a par allel slice. (d) The hyper determinant of a hypermatrix with two par allel slices pr oportional to each other is zer o. In particular , the hyperdeterminant of a hypermatrix with a zer o slice is zer o. For the rest of the paper we will only consider boundary format hypermatrices of three dimensions, that is, hypermatrices of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) for positive integers k 1 and k 2 . W e restrict our attention to these hypermatrices in order to make use of the next lemma, which, in conjunction with Corollary 2.4(d), will allo w us to classify which hypermatrices with restricted entries are nondegenerate. Lemma 2.5 ([Ait19, Lemma 2.3]) . Let M be a hypermatrix of boundary format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) over a ﬁeld F and let M 1 , M 2 , . . . , M k 1 +1 be the slices in the ﬁrst dir ection. Then M is nonde g enerate if and only if e very linear combination c 1 M 1 + c 2 M 2 + · · · + c k 1 +1 M k 1 +1 ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 5 with c 1 , c 2 , . . . , c k 1 +1 ∈ F not all zer o is full rank. W e no w take a brief detour from talking about hypermatrices to formalize some concepts from rook theory that we will be trying to generalize. For positiv e integers m and n we the deﬁne the m × n rectangle R m × n to be the subset { ( i, j ) : 1 ≤ i ≤ m and 1 ≤ j ≤ n } ⊆ Z > 0 × Z > 0 where we label ro ws of Z > 0 × Z > 0 top to bottom and columns left to right. F or instance, R 5 × 4 with this labeling scheme looks like (1 , 1) (1 , 2) (1 , 3) (1 , 4) (2 , 1) (2 , 2) (2 , 3) (2 , 4) (3 , 1) (3 , 2) (3 , 3) (3 , 4) (4 , 1) (4 , 2) (4 , 3) (4 , 4) (5 , 1) (5 , 2) (5 , 3) (5 , 4) . The main idea from rook theory we are trying to generalize is about rook placements on inte ger partition-shaped boar ds . W e formalize these in a complementary manner by looking at rectangles with an integer partition-shaped section remo ved. Deﬁnition 2.6. Let m and n be positiv e integers. An integer partition λ = ⟨ λ 1 , λ 2 , . . . , λ n ⟩ of format m × n is a list of nonneg ati ve inte gers λ i ≤ m such that λ i +1 ≤ λ i for i = 1 , 2 , . . . , n − 1 . For example, the integer partition λ = ⟨ 4 , 2 , 1 , 0 ⟩ is of format 4 × 4 . W e represent each partition by its Y oung diagram in transposed French notation; for example, the diagram of λ is . W e will write λ = ∅ to mean that λ is the empty partition λ 1 = λ 2 = · · · = λ n = 0 . For an inte ger partition λ of format m × n we deﬁne the board B λ as B λ = { ( i, j ) : 1 ≤ i ≤ m − λ j and 1 ≤ j ≤ n } ⊆ R m × n . If λ is clear from context we will often just write B . Intuitiv ely , B λ is the result of starting with the rectangle R m × n , lining up the bottom left hand corner of λ with R m × n , and then eliminating all the squares belonging to their union. For example, let λ = ⟨ 4 , 2 , 1 , 0 ⟩ as abov e. Then we vie w the board B λ as the white squares in the picture belo w: . Deﬁnition 2.7. A rook placement on B is a subset S of B such that for any ( i 1 , j 1 ) , ( i 2 , j 2 ) ∈ S we hav e i 1  = i 2 and j 1  = j 2 . 6 B. KOPR OWSKI AND J. B. LEWIS Intuiti vely we vie w a rook placement as a placement of non-attacking rooks in the squares of B or equiv alently a placement of rooks with at most one rook in each ro w and column. For instance, letting B λ be as abov e the rook placement { (2 , 2) , (1 , 3) , (4 , 4) } corresponds to r r r where r represents a rook. No w let us consider the set R n × n of rook placements of cardinality n on R n × n . Let S ∈ R n × n be some rook placement and let ( i j , j ) ∈ S be the pair with second coordinate j . Deﬁne f : R n × n → S n by letting f ( S ) be the permutation with f ( S )( j ) = i j for all j ∈ [ n ] . If we then let f − 1 : S n → R n × n be the function deﬁned by f − 1 ( σ ) = { ( σ ( j ) , j ) : j ∈ [ n ] } then we can see that f − 1 is the well deﬁned in v erse of f , so f is a bijection. Thus there is a natural bijectiv e correspondence between rook placements on an n × n board and permutations of S n . F or e xample, in the 4 × 4 case the rook placement r r r r corresponds to the permutation 4213 (written as a word in one-line notation). Deﬁnition 2.8. Let σ ∈ S n be a permutation. W e deﬁne the n × n permutation matrix m σ to be the matrix with ( m σ ) i,j = 1 if i = σ ( j ) and zero otherwise. W ith the preceding deﬁnition in mind it is clear that the bijection f between rook placements and permutations extends to a bijection between rook placements and permutation matrices. For instance, the matrix     0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0     corresponds to the permutation 4213 and pre vious rook placement. One will notice for a given rook placement S on B λ the following inequality holds: f ( S )( j ) ≤ n − λ j and therefore ( m f ( S ) ) i,j = 0 for all i > n − λ j . The main result from classical q -rook theory we are trying to generalize comes from looking at all matrices with entries ( i, j ) = 0 for all i > n − λ j ov er a ﬁnite ﬁeld F q . Deﬁnition 2.9. Let λ be an integer partition of format m × n and A an m × n matrix with entries in F . W e say that A respects λ if a i,j = 0 for all i > m − λ j for 1 ≤ j ≤ n . It immediately follo ws from the previous deﬁnition that if λ and µ are integer partitions such that µ i ≤ λ i for all 1 ≤ i ≤ n and a matrix A respects λ then A respects µ . Naturally , we will call µ a sub-partition of λ and write µ ⊆ λ . The follo wing result is an explicit product formula for the full-rank case of the e xpressions appear- ing in (1) (Haglund’ s theorem [Hag98, Thm. 1]), which relates the number of rook placements on B λ to the number of in v ertible matrices o ver F q that respect λ . Recall [ k ] q = 1 + q + · · · + q k − 1 for k ≥ 1 . ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 7 Theorem 2.10. Let λ be an inte ger partition with λ i ≤ n − i for i ∈ [ n ] . The number of placements of n r ooks on B λ is ( n − λ 1 )( n − 1 − λ 2 ) · · · (1 − λ n ) and the number of n × n in vertible matrices over F q that r espect λ is q ( n 2 ) ( q − 1) n [ n − λ 1 ] q [ n − 1 − λ 2 ] q · · · [1 − λ n ] q . These formulas are easy to prove recursi v ely , by b uilding the rook placement or matrix column by column from the left. In order to extend this connection to hypermatrices, we need an analogous notion of an integer partition for three dimensions. Deﬁnition 2.11. A plane partition P of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) is a collection ⟨ λ (1) , λ (2) , . . . , λ ( k 1 +1) ⟩ of integer partitions λ ( i ) of format ( k 1 + k 2 + 1) × ( k 2 + 1) such that λ ( i +1) ⊆ λ ( i ) for all 1 ≤ i ≤ k 1 . Analogous to the matrix case, we say a hypermatrix M of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) respects a plane partition P = ⟨ λ (1) , λ (2) , . . . , λ ( k 1 +1) ⟩ if each slice M i in the ﬁrst direction respects λ ( i ) for 1 ≤ i ≤ k 1 + 1 . For instance, if ⟨ λ (1) , λ (2) ⟩ is a plane partition of format 2 × 5 × 4 with λ (1) = ⟨ 4 , 2 , 1 , 0 ⟩ and λ (2) = ⟨ 3 , 2 , 1 , 0 ⟩ then any hypermatrix of the form       ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗       respects ⟨ λ (1) , λ (2) ⟩ . For con venience, if there exists a nondegenerate hypermatrix M that respects a plane partition P we say P allows nondegenerate hypermatrices. Similar to ho w we ha ve sub- partitions for integer partitions there is an analogous notion for plane partitions. Deﬁnition 2.12. Let P and P ′ be plane partitions of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) where P = ⟨ λ (1) , λ (2) , . . . , λ ( k 1 +1) ⟩ and P ′ = ⟨ µ (1) , µ (2) , . . . , µ ( k 1 +1) ⟩ . W e write P ′ ⊆ P and say P ′ is a sub-partition of P if µ ( i ) ⊆ λ ( i ) for all 1 ≤ i ≤ k 1 + 1 . The next lemma follo ws immediately from the deﬁnitions. Lemma 2.13. If P and P ′ ar e plane partitions with P ′ ⊆ P and P allows nonde gener ate hyperma- trices then P ′ allows nonde g enerate hypermatrices. Equi v alently if P ′ does not allow nondegenerate hypermatrices then P does not allow nondegen- erate hypermatrices. Lemma 2.14. Let P be a plane partition of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) with λ ( k 1 +1) k 2 +1 > 0 . Then P does not allow nonde g enerate hypermatrices. Pr oof. By Lemma 2.13 we may assume λ ( k 1 +1) k 2 +1 = 1 . By Deﬁnition 2.6, λ ( k 1 +1) k 2 +1 ≤ λ ( k 1 +1) i for 1 ≤ i ≤ k 2 + 1 . Furthermore, λ ( k 1 +1) ⊆ λ ( j ) for all 1 ≤ j ≤ k 1 + 1 and thus 1 = λ ( k 1 +1) k 2 +1 ≤ λ ( j ) i for all 1 ≤ i ≤ k 2 + 1 and 1 ≤ j ≤ k 1 + 1 . Let M = ( a i 1 ,i 2 ,i 3 ) be a hypermatrix that respects P , so by deﬁnition a i 1 ,i 2 ,i 3 = 0 whenev er i 2 > k 1 + k 2 + 1 − λ ( i 1 ) i 3 . Therefore a i 1 , ( k 1 + k 2 +1) ,i 3 = 0 for all 1 ≤ i 1 ≤ k 1 + 1 and 1 ≤ i 3 ≤ k 2 + 1 . Howe ver , this is the same as saying the ( k 1 + k 2 + 1) slice in the 2 nd direction is zero and by Corollary 2.4(d) we conclude that M is degenerate. □ By Lemma 2.5 we get the follo wing corollary . 8 B. KOPR OWSKI AND J. B. LEWIS Corollary 2.15. Let P be a plane partition of format ( k 1 +1) × ( k 1 + k 2 +1) × ( k 2 +1) with λ ( k 1 +1) k 2 +1 > 0 and M a hypermatrix that respects P with slices M 1 , M 2 , . . . , M k 1 +1 in the ﬁrst dir ection. Ther e exists some c 1 , c 2 , . . . , c k 1 +1 ∈ F q , not all zer o, such that c 1 M 1 + c 2 M 2 + · · · + c k 1 +1 M k 1 +1 has less than full rank. It appears now that we have deﬁned an analogous version of the partition-shaped restrictions on matrices for hypermatrices. F or this to be truly analogous in the q -rook counting sense there is one property we would like to ensure holds. Let us go back to classical q -rook theory for a moment and ﬁx a format n × n . By basic linear algebra the partition δ = ⟨ n − 1 , n − 2 , . . . , 1 , 0 ⟩ is maximal such that there exists an in vertible matrix which respects δ . In the 4 × 4 case all matrices of form     ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 0 ∗ ∗ 0 0 0 ∗     respect δ , and in particular the standard 4 × 4 identity matrix respects δ . W e no w deﬁne the corre- sponding objects in three dimensions, and sho w that the analogous result holds for them. Deﬁnition 2.16. Fix a format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) . W e denote by ∆ k 1 ,k 2 the plane partition ⟨ λ (1) , . . . , λ ( k 1 +1) ⟩ where λ ( i ) j = ( k 1 + k 2 + 1) − ( i + j − 1) for 1 ≤ i ≤ k 1 + 1 and 1 ≤ j ≤ k 2 + 1 . Further , we denote by E the hypermatrix with entry E i 1 ,i 2 ,i 3 equal to 1 when i 2 = i 1 + i 3 − 1 and 0 otherwise. Lemma 2.17 ([GKZ94, Lemma 3.4]) . F or any format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) , the hypermatrix E is nonde generate . Theorem 2.18. Let ∆ k 1 ,k 2 be as in Deﬁnition 2.16. Then ∆ k 1 ,k 2 is the unique maximal plane partition with r espect to the or dering in Deﬁnition 2.12 allowing nonde gener ate hypermatrices. Pr oof. By deﬁnition ∆ k 1 ,k 2 allo ws E which is nondegenerate by Lemma 2.17. Now we show if Q ⊆ ∆ k 1 ,k 2 then Q does not allo w nondegenerate h ypermatrices. If Q = ⟨ µ (1) , . . . , µ ( k 1 +1) ⟩ is some plane partition of format ( k 1 + 1) × ( k 1 + k 2 + 1) × ( k 2 + 1) such that Q ⊆ ∆ k 1 ,k 2 , then there exists some i, j > 0 such that µ ( i ) j > ( k 1 + k 2 + 1) − ( i + j − 1) . Let P = ⟨ λ (1) , . . . , λ ( i ) , ∅ , . . . , ∅⟩ where λ (1) = · · · = λ ( i ) and λ ( i ) 1 = · · · = λ ( i ) j = ( k 1 + k 2 + 1) − ( i + j − 1) + 1 and λ ( i ) j +1 = · · · = λ ( i ) k 2 +1 = 0 , so P ⊆ Q . By Lemma 2.13 to prove Q does not allow nonde generate hypermatrices it sufﬁces to sho w P does not allo w nondegenerate hypermatrices. Let M = ( a i 1 ,i 2 ,i 3 ) be a hypermatrix which respects P . Let N be the sub-hypermatrix of format [( i − 1) + 1] × [( i − 1) + ( j − 1) + 1] × [( j − 1) + 1] that consists of the ﬁrst i + j − 1 ro ws and j columns of the ﬁrst i slices of M in the ﬁrst direction. So, N = ( a i 1 ,i 2 ,i 3 ) where 1 ≤ i 1 ≤ i , 1 ≤ i 2 ≤ i + j − 1 , and 1 ≤ i 3 ≤ j . By the assumption that M respects P , we kno w ( a i 1 ,i 2 ,i 3 ) = 0 for 1 ≤ i 1 ≤ i , 1 ≤ i 3 ≤ j , and i 2 > ( k 1 + k 2 + 1) − [( k 1 + k 2 + 1) − ( i + j − 1) + 1] = i + j − 2 . Therefore a i 1 , ( i + j − 1) ,i 3 = 0 for all 1 ≤ i 1 ≤ i and 1 ≤ i 3 ≤ j , or equiv alently the i + j − 1 slice of N in the second direction is zero. By Corollary 2.15 there e xists c 1 , c 2 , . . . , c i ∈ F q not all zero such that c 1 N 1 + c 2 N 2 + · · · + c i N i ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 9 is less than full rank. Let us denote the result of this linear combination as the ( i + j − 1) × j matrix of column vectors [ v 1 | v 2 | · · · | v j ] . No w we tak e linear combination c 1 M 1 + · · · + c i M i + 0 M i +1 + · · · + 0 M k 1 +1 of slices of M in the ﬁrst direction and let us denote this sum as the ( k 1 + k 2 + 1) × ( k 2 + 1) matrix of column vectors [ u 1 | u 2 | · · · | u k 2 +1 ] . The ﬁrst j columns of this matrix are      v 1 0 . . . 0      ,      v 2 0 . . . 0      , . . . ,      v j 0 . . . 0      which are linearly dependent. Thus the ﬁrst j columns of [ u 1 | u 2 | · · · | u k 2 +1 ] ha ve rank less than j and thus the matrix has rank less than k 2 + 1 . It follows by Lemma 2.5 that M is degenerate. Thus P does not allo w nondegenerate hypermatrices and the theorem follo ws. □ 3. M A I N C O N J E C T U R E In this section we are only concerned with hypermatrices of format 2 × ( k + 1) × k . When it is clear from context we will refer to the plane partition ∆ 1 , ( k − 1) as ∆ and an arbitrary plane partition ⟨ λ (1) , λ (2) ⟩ as ⟨ λ, µ ⟩ where λ and µ are integer partitions of format ( k + 1) × k . Our main goal in this section is to introduce and study the following conjectural enumeration of nondegenerate hypermatrices of format 2 × ( k + 1) × k that a void a particular plane partition P ⊆ ∆ . Conjecture 3.1. F ix a format 2 × ( k + 1) × k and let P = ⟨ λ, µ ⟩ be a plane partition such that P ⊆ ∆ . The number of nondeg enerate 2 × ( k + 1) × k hypermatrices over F q which r espect P is q k 2 ( q − 1) 2 k [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q · [ k − µ 1 ] q [( k − 1) − µ 2 ] q · · · [1 − µ k ] q . Although we will not be able to prove this conjecture for all P , there are large families of P for which we will show it holds. But, before we get into hypermatrix counting we need to establish a fe w crucial deﬁnitions and a theorem from Aitken that will be the main tool in counting such objects. Deﬁnition 3.2. Fix a ﬁeld F and a format 2 × ( k + 1) × k . Let M k be the set of nondegenerate hypermatrices ov er F of format 2 × ( k + 1) × k . Deﬁne the group G = GL k +1 ( F ) × GL k ( F ) / N where N is the subgroup of GL k +1 ( F ) × GL k ( F ) of ordered pairs ( cI k +1 , c − 1 I k ) for c ∈ F × . W e need to be careful about what we say next because this quotient group G does not automatically inherit a group action on M k from GL 2 ( F ) × GL k +1 ( F ) × GL k ( F ) ; howe v er in this case we claim that it does. First, notice GL k +1 ( F ) × GL k ( F ) inherits the group action of GL 2 ( F ) × GL k +1 ( F ) × GL k ( F ) by identifying GL k +1 ( F ) × GL k ( F ) with the subgroup 1 × GL k +1 ( F ) × GL k ( F ) where 1 is the trivial subgroup of GL 2 ( F ) . Second, ev ery element of N commutes with ev ery element of GL k +1 ( F ) × GL k ( F ) , so N is in the center of GL k +1 ( F ) × GL k ( F ) . Therefore N is normal and the group G is well deﬁned. Notice also that ev ery element of N acts trivially on the set M k . Now choose an element ( g 1 , g 2 ) N ∈ G , for ( g 1 , g 2 ) ∈ GL k +1 ( F ) × GL k ( F ) , and a coset representati v e ( g 1 , g 2 ) · ( cI k +1 , c − 1 I k ) . W e deﬁne the action of ( g 1 , g 2 ) N on M k by the action the representative ( g 1 , g 2 ) · ( cI k +1 , c − 1 I k ) inherits from GL k +1 ( F ) × GL k ( F ) . This is well deﬁned since for an y other coset representati ve and M ∈ M k ( g 1 , g 2 ) · ( bI k +1 , b − 1 I k ) ◦ M = ( g 1 , g 2 ) ◦ M = ( g 1 , g 2 ) · ( cI k +1 , c − 1 I k ) ◦ M since the elements of N act tri vially on M k . Therefore the action of GL k +1 ( F ) × GL k ( F ) on M k extends to a well deﬁned action of G . Theorem 3.3 ([Ait19, Thm. 2.2]) . The action of G on M k is a fr ee, tr ansitive action. 10 B. KOPR OWSKI AND J. B. LEWIS It follo ws immediately that there is a bijection between G and M k and we get the follo wing corol- lary , which is also the case λ = µ = ∅ of Conjecture 3.1. Corollary 3.4 ([Ait19, Prop. 3.1]) . The number of nonde gener ate 2 × ( k + 1) × k hypermatrices o ver F q is q k 2 ( q − 1) 2 k [ k + 1]! q [ k ]! q . The previous corollary follo ws by computing the size of G but also leads to a general idea of how to count nondegenerate hypermatrices using the action of G . Speciﬁcally , if we want to count the nondegenerate hypermatrices that respect a plane partition P ⊆ ∆ we ﬁrst look at the set S = { ( g 1 , g 2 ) ∈ GL k +1 ( F q ) × GL k ( F q ) : ( g 1 , g 2 ) ◦ E respects P } , where E is the hypermatrix in Deﬁnition 2.16. Then the number of nondegenerate hypermatrices that respect P is | S | / | N | = | S | / ( q − 1) . W e will apply this idea numerous times when counting families of nondegenerate hypermatrices. No w we will prov e the following two technical lemmas which will allow us to extend the special case abov e to the much broader family of plane partitions with µ = ∅ and λ arbitrary . Lemma 3.5. Let { v 1 , v 2 , . . . , v n } be a linearly independent collection of vectors in F k +1 q for 1 ≤ n ≤ k + 1 . F or a vector w ∈ F k +1 q let w · v be the standar d dot pr oduct. Ther e are q k +1 − n vectors x ∈ F k +1 q such that x · v i = 0 for 1 ≤ i ≤ n . Pr oof. For a vector v ∈ F k +1 q deﬁne v T to be the transpose of v . W e deﬁne the linear transformation T A : F k +1 q → F k +1 q by the matrix of ro w vectors A =           — v T 1 — . . . — v T n — — 0 — . . . — 0 —           where 0 represents the zero vector . For a v ector x ∈ F k +1 q one can see T A ( x ) = Ax =  x · v 1 x · v 2 · · · x · v n 0 · · · 0  T and notice that Ker( T A ) is the set of vectors we are trying to count. By the rank-nullity theorem dim Ker( T A ) = k + 1 − dim Im( T A ) and we claim that dim Im( T A ) = n . T o see this recall dim Im( T A ) is equal to the dimension of the row space of A . Clearly the dimension of the ro w space is n , so dim Im( T A ) = n and dim Ker( T A ) = k + 1 − n . Therefore | Ker( T A ) | = q k +1 − n as claimed. □ Lemma 3.6. Let λ be an inte ger partition of format ( k + 1) × k with λ i ≤ k + 1 − i and λ T = ⟨ λ T 1 , λ T 2 , . . . , λ T k +1 ⟩ be the transpose of λ . Then [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q = [ k + 1 − λ T 1 ] q [ k − λ T 2 ] q · · · [1 − λ T k +1 ] q . Pr oof. T o pro ve this we count nonde generate ( k + 1) × ( k + 1) matrices that respect λ in two different ways. Fix a prime power q . By Theorem 2.10 there are q ( k +1 2 ) ( q − 1) k +1 [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q such matrices ov er F q , and we hav e dropped the factor [1 − λ k +1 ] q since by assumption λ k +1 = 0 . Next, we count the same objects e xcept ﬁrst we conjugate by the anti-diagonal matrix of all ones, and ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 11 then take the transpose. It is easy to see this map is a bijection between ( k + 1) × ( k + 1) in v ertible matrices that respect λ and those that respect λ T . Thus, by Theorem 2.10 there are q ( k +1 2 ) ( q − 1) k +1 [ k + 1 − λ T 1 ] q [ k − λ T 2 ] q · · · [1 − λ T k +1 ] q such matrices. W e have counted the same object two different ways so the displayed equations are equal. Since the y are equal for each prime po wer q , they are equal as polynomials, and the statement follo ws after canceling po wers of q and q − 1 . □ 3.1. The case µ = ∅ . Our next result sho ws that Conjecture 3.1 holds for all P ⊆ ∆ with µ = ∅ . Theorem 3.7. Let P ⊆ ∆ with µ = ∅ . The number of nondeg ener ate hypermatrices which respect P is q k 2 ( q − 1) 2 k [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q · [ k ]! q . Pr oof. By Theorem 3.3 e v ery nonde generate hypermatrix can be written as ( g 1 , g 2 ) ◦ E for ( g 1 , g 2 ) ∈ GL k +1 ( F q ) × GL k ( F q ) . W e count the number of pairs ( g 1 , g 2 ) such that ( g 1 , g 2 ) ◦ E respects P . In particular , we begin by ﬁxing a g 2 ∈ GL k ( F q ) and then compute the number of g 1 such that ( g 1 , g 2 ) ◦ E respects P . Fix a format 2 × ( k + 1) × k and let P ⊆ ∆ be a plane partition with µ = ∅ . For some g 2 ∈ GL k ( F q ) let M be the hypermatrix M = (1 , g 2 ) ◦ E . W e denote the entries of M as M i 1 ,i 2 ,i 3 where i 1 ∈ [2] , i 2 ∈ [ k + 1] , and i 3 ∈ [ k ] . Let g 1 ∈ GL k +1 ( F q ) such that ( g 1 , 1) ◦ M respects P and notice ( g 1 , 1) ◦ M = ( g 1 , g 2 ) ◦ E . W e will denote g 1 by the matrix of ro w vectors g 1 =    — v 1 — . . . — v k +1 —    and by assumption g 1 ∈ GL k +1 ( F q ) the collection { v 1 , v 2 , . . . , v k +1 } is a linearly independent set of vectors. Let us represent the hypermatrix M as the pair of slices in the ﬁrst direction ( M 1 , M 2 ) and we will refer to each as a matrix of column vectors M 1 =   a 1 , 1 a 1 , 2 . . . a 1 ,k   , M 2 =   a 2 , 1 a 2 , 2 . . . a 2 ,k   . By Theorem 3.3 the hypermatrix M is nondegenerate, so by Lemma 2.5 the collections of v ectors { a 1 , 1 , a 1 , 2 , . . . , a 1 ,k } and { a 2 , 1 , a 2 , 2 , . . . , a 2 ,k } are both linearly independent. By the deﬁnition of the action, entry ( i 1 , i 2 , i 3 ) of ( g 1 , 1) ◦ M is equi v alent to v i 2 · a i 1 ,i 3 . Then by the assumption ( g 1 , 1) ◦ M respects P we ha ve v i 2 · a 1 ,i 3 = 0 for all i 2 > k + 1 − λ i 3 with i 3 ∈ [ k ] or equi v alently v i 2 · a 1 ,i 3 = 0 for 1 ≤ i 3 ≤ λ T k +2 − i 2 with i 2 ∈ [ k + 1] . Howe v er , since µ = ∅ these are the only conditions that the collection of vectors { v 1 , v 2 , . . . , v k +1 } must satisfy in order for ( g 1 , 1) ◦ M to respect P . Now we count ho w many choices of v ectors v 1 , v 2 , . . . , v k +1 we hav e thereby counting the number of g 1 . For i ∈ [ k + 1] deﬁne S i = { x ∈ F k +1 q | x · a 1 ,j = 0 for all 1 ≤ j ≤ λ T k +2 − i } , that is S i is the set of all vectors x ∈ F k +1 q such that the dot product of x with the ﬁrst λ T k +2 − i col- umn vectors of M 1 is zero. Notice that S i has the property that if { x 1 , x 2 , . . . , x n } ⊆ S i then Span { x 1 , x 2 , . . . , x n } ⊆ S i . By the fact { a 1 , 1 , , a 1 , 2 , . . . , a 1 ,k } is a linearly independent set of v ec- tors we can use Lemma 3.5 and calculate | S i | = q k +1 − λ T k +2 − i . Notice λ T k +2 − i ≤ λ T k +2 − ( i +1) , so S k +1 ⊆ S k ⊆ · · · ⊆ S 1 . By assumption ( g 1 , 1) ◦ M respects P we ha ve v k +1 ∈ S k +1 and since v k +1  = 0 there are q k +1 − λ T 1 − 1 choices for v k +1 . Generally , for 1 ≤ i ≤ k we know v k +1 − i ∈ S k +1 − i and Span { v k +1 , v k , . . . , v k +1 − ( i − 1) } ⊆ S k +1 − i . By the linear independence of { v k +1 , . . . , v k +1 − i } we have v k +1 − i ∈ S k +1 − i \ Span { v k +1 , v k , . . . , v k +1 − ( i − 1) } so there are q k +1 − λ T i +1 − q i choices for v k +1 − i . Therefore the number of choices of v k +1 , v k , . . . , v 1 is ( q k +1 − λ T 1 − 1)( q k +1 − λ T 2 − q ) · · · ( q k +1 − λ T k +1 − q k ) 12 B. KOPR OWSKI AND J. B. LEWIS which is also the number of choices g 1 such that ( g 1 , 1) ◦ M respects P . This factors as q ( k +1 2 ) ( q + 1) k +1 [ k + 1 − λ T 1 ] q [ k − λ T 2 ] q · · · [2 − λ T k ] q [1 − λ T k +1 ] q and by Lemma 3.6 this is equal to q ( k +1 2 ) ( q + 1) k +1 [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q . Since we assumed g 2 ∈ GL k ( F q ) was arbitrary there are q k 2 ( q − 1) 2 k +1 [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q · [ k ]! q pairs ( g 1 , g 2 ) ∈ GL k +1 ( F q ) × GL k ( F q ) such that ( g 1 , g 2 ) · E respects P . Similar to the counting in Corollary 3.4 we conclude there are q k 2 ( q − 1) 2 k [ k + 1 − λ 1 ] q [ k − λ 2 ] q · · · [2 − λ k ] q · [ k ]! q nondegenerate hypermatrices that respect P . □ 3.2. The case P = ∆ . W e giv e one more family of plane partitions for which the conjecture holds. Theorem 3.8. The number of nonde generate hypermatrices r especting ∆ is q k 2 ( q − 1) 2 k . Pr oof. Let L k , U k +1 respecti vely denote the sets of lo wer -triangular k × k and upper -triangular ( k + 1) × ( k + 1) in vertible matrices, let S 1 denote the set of nondegenerate 2 × ( k + 1) × k hypermatrices that respect ∆ , and let S 2 denote the set of 2 × ( k + 1) × k hypermatrices M for which M i,j,k  = 0 if j = i + k − 1 and for which M i,j,k = 0 if j > i + k − 1 . W e claim that ( U k +1 , L k ) ◦ E = S 1 = S 2 . W e proceed by proving two containments and then showing the largest and smallest of the three sets hav e the same size. It is easy to see from the deﬁnition of the action of matrices on hypermatrices that if A is upper- triangular and B is lo wer -triangular then ( A, B ) ◦ E respects ∆ . Since E is nondegenerate, it follo ws immediately from Proposition 2.3 that ( U k +1 , L k ) ◦ E ⊆ S 1 . By Theorem 2.18, if a hypermatrix M respects ∆ and M 1 ,i 3 ,i 3 = 0 or M 2 ,i 3 +1 ,i 3 = 0 for i 3 ∈ [ k ] then M is degenerate; consequently S 1 ⊆ S 2 . Finally , by directly counting the number of choices for each entry we observe that | S 2 | = q ( k 2 ) + ( k +1 2 ) ( q − 1) 2 k = q k 2 ( q − 1) 2 k , whereas since both U k +1 and L k contain the scalar matrices, we hav e by Theorem 3.3 that | ( U k +1 , L k ) ◦ E | = 1 q − 1 | U k +1 | · | L k | = q k 2 ( q − 1) 2 k , as well. Thus in fact all three sets are equal, and ha ve the claimed cardinality . □ 3.3. The case k = 2 . Although we do not hav e a general theorem to count the number of nonde- generate hypermatrices for an arbitrary plane partition with µ  = ∅ many of the methods we hav e de veloped so far can be applied to compute concrete cases. T o illustrate the use of these techniques we will compute the number of nondegenerate hypermatrices in the 2 × 3 × 2 case with P  = ∆ and µ  = ∅ . In this case there are three plane partitions which satisfy these properties: (1) P 1 with λ = ⟨ 1 , 0 ⟩ and µ = ⟨ 1 , 0 ⟩ , (2) P 2 with λ = ⟨ 2 , 0 ⟩ and µ = ⟨ 1 , 0 ⟩ , (3) P 3 with λ = ⟨ 1 , 1 ⟩ and µ = ⟨ 1 , 0 ⟩ . Although this case is small the general strategies used to compute the number of nondegenerate hypermatrices that respect P i for i = 1 , 2 , 3 can be used in larger cases, especially the method used for i = 3 . Also, the number of plane partitions with P  = ∆ and µ  = ∅ grows rapidly , for example in the 2 × 4 × 3 case there are 40 such P and we would rather not put the reader through all of that. ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 13 Let us start with P 1 and look at the hypermatrix (1 , g 2 ) ◦ E for some g 2 ∈ GL 2 ( F q ) . Let g 2 =  a b c d  and recall there are q ( q − 1) 2 [2] q such g 2 . By deﬁnition of the action (1 , g 2 ) · E =   a c b d 0 0     0 0 a c b d   and recall by Proposition 2.3 this hypermatrix is nonde generate. Therefore by Lemma 2.5 the vectors   a b 0   and   0 a b   are linearly independent. Now let us take some g 1 ∈ GL 3 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 1 . W e will think of g 1 as the matrix of row vectors   — r 1 — — r 2 — — r 3 —   . By assumption ( g 1 , g 2 ) ◦ E respects P we have r 3 ·   a b 0   = r 3 ·   0 a b   = 0 and by Lemma 3.5 there are q − 1 choices for r 3 . The second ro w , r 2 , has no restrictions other than it must be linearly independent from r 3 , so there are q 3 − q choices for r 2 . Similarly , r 1 has no restrictions other than being linearly independent from r 2 and r 3 so there are q 3 − q 2 choices of r 1 . Thus the number of ( g 1 , g 2 ) such that ( g 1 , g 2 ) · E respects P 1 is ( q − 1)( q 3 − q )( q 3 − q 2 ) · q ( q − 1) 2 [2] q = q 4 ( q − 1) 5 [2] 2 q . Therefore the number of nondegenerate hypermatrices which respect P 1 is q 4 ( q − 1) 4 [2] 2 q . W e play the same game on P 2 , choosing any g 2 ∈ GL 2 ( F q ) which we represent with the same matrix as before. Again we let g 1 ∈ GL 3 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 2 and we represent g 1 as the matrix of row vectors abov e. As before we must have r 3 ·   a b 0   = 0 and r 3 ·   0 a b   = 0 so there are q − 1 choices for r 3 . No w we must hav e r 2 ·   a b 0   = 0 and by assumption g 1 is in v ertible r 2 / ∈ Span { r 3 } . Ho we v er e very scalar multiple of r 3 is also a choice for r 2 since ( cr 3 ) ·   a b 0   = c   r 3 ·   a b 0     = 0 and so there are q 2 − q choices for r 2 . No w we need to choose r 1 and the only restriction we hav e comes from the assumption g 1 is in v ertible so r 1 / ∈ Span { r 3 , r 2 } and there are q 3 − q 2 choices for r 1 . Thus the number of ( g 1 , g 2 ) ∈ GL 3 ( F q ) × GL 2 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 2 is ( q − 1)( q 2 − q )( q 3 − q 2 ) q ( q − 1) 2 [2] q = q 4 ( q − 1) 5 [2] q and the number of nondegenerate hypermatrices that respect P 2 is q 4 ( q − 1) 4 [2] q . In the pre vious two cases, we were able to get by with the same techniques as in the case µ = ∅ of Theorem 3.7. F or P 3 something genuinely new happens; that is for ev ery choice of g 2 ∈ GL 2 ( F q ) it is not true that there are the same number of g 1 ∈ GL 3 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 3 . Instead of assuming g 2 ∈ GL 2 ( F q ) is arbitrary we will consider the follo wing two cases: g 2 =  a 0 c d  and 14 B. KOPR OWSKI AND J. B. LEWIS g 2 =  a x c d  where x ∈ F × q . In the ﬁrst case (1 , g 2 ) ◦ E =   a c 0 d 0 0     0 0 a c 0 d   and now let us count the g 1 ∈ GL 3 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 3 . Recall the diagonal of an in v ertible upper triangular matrix is nonzero and since g 2 is in v ertible, a, d  = 0 and c is free so there are q ( q − 1) 2 such g 2 . Let the third ro w of g 1 be  w y z  . W e must ha ve  w y z  ·   a 0 0   = 0 so w = 0 and similarly  w y z  ·   0 a 0   = 0 so y = 0 . By assumption g 1 is in v ertible we know z  = 0 so z ∈ F × q . F or any such z ,  0 0 z  ·   a 0 0   =  0 0 z  ·   0 a 0   =  0 0 z  ·   c d 0   = 0 and thus there are q − 1 choices for  w y z  . Since those are all the restrictions on g 1 the number of such matrices is ( q − 1)( q 3 − q )( q 3 − q 2 ) . Let us see what happens when we let g 2 =  a x c d  with x ∈ F × q . By deﬁnition of the action we hav e (1 , g 2 ) · E =   a c x d 0 0     0 0 a c x d   and let us assume there exists some g 1 ∈ GL 3 such that ( g 1 , g 2 ) ◦ E respects P 3 . Let us represent g 1 as the matrix of row vectors   — r 1 — — r 2 — — r 3 —   . By the assumption that ( g 1 , g 2 ) ◦ E respects P 3 we hav e r 3 ·   a x 0   = r 3 ·   c d 0   = 0 . Since   a x 0   and   c d 0   are linearly independent by Lemma 3.5 there are q − 1 possible choices for r 3 . W e can see these are the v ectors  0 0 z  with z ∈ F × q . Ho we v er , by assumption ( g 1 , g 2 ) ◦ E respects P 3 we also hav e r 3 ·   0 a x   = z x = 0 which is a contradiction since we assumed both z , x ∈ F × q . Thus for any choice of g 2 =  a x c d  with x ∈ F × q there are no ( g 1 , g 2 ) ∈ GL 3 ( F q ) × GL 2 ( F q ) such that ( g 1 , g 2 ) ◦ E respects P 3 . Thus the number of elements ( g 1 , g 2 ) such that ( g 1 , g 2 ) ◦ E respects P 3 is ( q − 1)( q 3 − q )( q 3 − q 2 ) q ( q − 1) 2 = q 4 ( q − 1) 5 [2] q and the number of nondegenerate hypermatrices that respect P 3 is q 4 ( q − 1) 4 [2] q . ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 15 4. H Y P E R R O O K P L AC E M E N T S A N D A W E A K F O R M O F T H E M A I N C O N J E C T U R E W e now switch gears and look at the “moral” e vidence supporting the conjecture. W e begin by deﬁning a natural generalization of rook placements that we call hyperr ook placements . The main result of this section is the follo wing theorem, which veriﬁes sev eral nontri vial predictions of Conjec- ture 3.1, namely , that the number of nonde generate 2 × ( k + 1) × k hypermatrices over F q with support on a plane partition is a polynomial function of q that has (up to a power of q − 1 ) nonne gati ve integer coef ﬁcients, and furthermore pro vides a combinatorial interpretation for the v alue of this polynomial at q = 1 . Theorem 4.1. Let P ⊆ ∆ . Ther e is a polynomial f with nonnegative inte ger coefﬁcients suc h that f (1) = ( k + 1 − λ 1 )( k − λ 2 ) · · · (2 − λ k ) · ( k − µ 1 )(( k − 1) − µ 2 ) · · · (1 − µ k ) is the number of hyperr ook placements that r espect P and, for every prime power q , the number of nonde g enerate hypermatrices that r espect P is q k 2 ( q − 1) 2 k · f ( q ) . 4.1. Hyperrook placements. W e begin by deﬁning the combinatorial model we use for higher- dimensional rook placements. Deﬁnition 4.2. Let σ ∈ S k +1 and π ∈ S k and m σ , m π be their corresponding permutation matrices in GL k +1 ( F ) and GL k ( F ) , respecti vely . A 2 × ( k + 1) × k hyperr ook placement is an y hypermatrix of the form ( m σ , m π ) ◦ E where E is the hypermatrix deﬁned in Deﬁ nition 2.16. Example 4.3. The hyperrook placement ( m 4213 , m 312 ) ◦ E is gi v en by ( m 4213 , m 312 ) ◦ E = ( m 4213 , 1) ◦   1 ,   0 1 0 0 0 1 1 0 0     ◦         1 0 0 0 1 0 0 0 1 0 0 0         0 0 0 1 0 0 0 1 0 0 0 1         =         0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0     , I     ◦         0 0 1 1 0 0 0 1 0 0 0 0         0 0 0 0 0 1 1 0 0 0 1 0         =     0 1 0 1 0 0 0 0 0 0 0 1         1 0 0 0 0 1 0 1 0 0 0 0     . The question of whether there is a more intrinsic deﬁnition of hyperrook placements is raised in Section 5.3 belo w . W e now build up to showing that the number of hyperrook placements that respect a plane partition P ⊆ ∆ corresponds to the conjectured number of nondegenerate hypermatrices that respect P as in Conjecture 3.1. In order to do this, we need to establish some deﬁnitions. Deﬁnition 4.4. Giv en ( σ, π ) ∈ S k +1 × S k , let π be the permutation in S k +1 with π ( i ) = π ( i ) for i ∈ [ k ] and π ( k + 1) = k + 1 . Deﬁne w , c ∈ S k +1 by w = σ · π and c = π − 1 (1 2 · · · k k + 1) π . Deﬁnition 4.5. Let C k +1 ⊆ S k +1 × S k +1 be the set of all ordered pairs ( α, β ) where α is an arbitrary element of S k +1 and β is a ( k + 1) -cycle. It turns out that the map of Deﬁnition 4.4 is actually a bijection, as the follo wing proposition shows. Proposition 4.6. The function S k +1 × S k → C k +1 that maps ( σ , π ) 7→ ( w , c ) as in Deﬁnition 4.4 is a bijection. 16 B. KOPR OWSKI AND J. B. LEWIS Pr oof. First we show the map is injecti v e. For ( σ 1 , π 1 ) , ( σ 2 , π 2 ) ∈ S k +1 × S k , suppose π 1 − 1 (1 2 · · · k + 1) π 1 = π 2 − 1 (1 2 · · · k + 1) π 2 and σ 1 · π 1 = σ 2 · π 2 . Then π 2 π 1 − 1 (1 2 · · · k + 1)( π 2 π 1 − 1 ) − 1 = (1 2 · · · k + 1) . Therefore  π 2 π 1 − 1 (1) π 2 π 1 − 1 (2) · · · π 2 π 1 − 1 ( k ) π 2 − 1 π 1 ( k + 1)  = (1 2 · · · k k + 1) . Notice π 1 − 1 ( k + 1) = π 2 ( k + 1) = k + 1 so π 2 π 1 − 1 ( k + 1) = k + 1 . Therefore we must have π 2 π 1 − 1 ( i ) = i for all i ∈ [ k ] . But then π 1 ( i ) = π 1 ( i ) = π 2 ( i ) = π 2 ( i ) for all i ∈ [ k ] , and so π 1 = π 2 . Consequently , π 1 = π 2 and by the assumption σ 1 · π 1 = σ 2 · π 2 we get σ 1 = σ 2 . Therefore ( σ 1 , π 1 ) = ( σ 2 , π 2 ) and the map is injectiv e. Since the domain and codomain both have size ( k + 1)! · k ! , the map is a bijection, as claimed. □ Consider the pair ( σ, π ) = (4213 , 231) . W e compute w = 4213 · 2314 = 2143 , c = 3124 · (1 2 3 4) · 2314 = (3 1 2 4) , and w c = 2143 · (3 1 2 4) = 1324 . Comparing this to the hyperrook placement ( m 4213 , m 312 ) ◦ E in Example 4.3, we see the 1 s are exactly in positions (1 , w ( j ) , j ) and (2 , w c ( j ) , j ) for j = 1 , 2 , 3 . The ne xt lemma sho ws this is not a coincidence. Lemma 4.7. Let ( σ, π ) ∈ S k +1 × S k and ( w , c ) be deﬁned as above. Let M be the hypermatrix of format 2 × ( k + 1) × k with M 1 ,i,j = 1 if i = w ( j ) and M 2 ,i,j = 1 if i = w c ( j ) for j ∈ [ k ] , and all other entries zer o. Then ( m σ , m π − 1 ) ◦ E = M . Pr oof. Denote by A 1 , . . . , A k the slices of E = ( e i 1 ,i 2 ,i 3 ) in the third direction and by B 1 , . . . , B k the slices of (1 , m π − 1 ) ◦ E in the third direction. By the deﬁnition of the group action, we have for n = 1 , . . . , k that B n = ( m π − 1 ) n, 1 · A 1 + ( m π − 1 ) n, 2 · A 2 + · · · + ( m π − 1 ) n,k · A k = A π ( n ) . Thus, the ( i 1 , i 2 , i 3 ) -entry of (1 , m π − 1 ) ◦ E is e i 1 ,i 2 ,π ( i 3 ) . Considering in the same way the action of ( m σ , 1) on (1 , m π − 1 ) ◦ E by permuting slices in the second direction, we conclude that the ( i 1 , i 2 , i 3 ) - entry of ( m σ , m π − 1 ) ◦ E is e i 1 ,σ − 1 ( i 2 ) ,π ( i 3 ) . Since e 1 ,i,j is 1 if i = j and is 0 otherwise, and e 2 ,i,j is 1 if i = j + 1 and is 0 otherwise, we conclude that the (1 , i 2 , i 3 ) -entry of ( m σ , m π − 1 ) ◦ E is 1 when π ( i 3 ) = σ − 1 ( i 2 ) and the (2 , i 2 , i 3 ) -entry is 1 when π ( i 3 ) + 1 = σ − 1 ( i 2 ) . Equi v alently , the (1 , i 2 , i 3 ) entry is 1 when w ( i 3 ) = σ ( π ( i 3 )) = i 2 for 1 ≤ i 3 ≤ k , and the (2 , i 2 , i 3 ) -entry is 1 when σ ( π ( i 3 ) + 1) = i 2 for 1 ≤ i 3 ≤ k . Notice π ( i 3 ) + 1 = (1 2 · · · k + 1) π ( i 3 ) for all 1 ≤ i 3 ≤ k . Therefore σ ( π ( i 3 ) + 1) = σ (1 2 · · · k + 1) π ( i 3 ) = σ π π − 1 (1 2 · · · k + 1) π ( i 3 ) = w c ( i 3 ) for all 1 ≤ i 3 ≤ k . Thus the (2 , i 2 , i 3 ) -entry of ( m σ , m π − 1 ) ◦ E is 1 precisely when w c ( i 3 ) = i 2 for 1 ≤ i 3 ≤ k . Since all other entries of ( m σ , m π − 1 ) ◦ E are 0 , we conclude that ( m σ , m π − 1 ) ◦ E = M , as claimed. □ Corollary 4.8. Let P ⊆ ∆ be a plane partition of format 2 × ( k + 1) × k . A hyperr ook placement ( m σ , m π − 1 ) ◦ E respects P if and only if the following two conditions ar e satisﬁed for all 1 ≤ j ≤ k : (1) w ( j ) ≤ k + 1 − λ j and (2) w c ( j ) ≤ k + 1 − µ j , wher e w and c ar e as in Deﬁnition 4.4. Pr oof. Let ( m σ , m π − 1 ) ◦ E be a hyperrook placement. By deﬁnition, M := ( m σ , m π − 1 ) ◦ E respects P if and only if M 1 ,i,j = 0 for all i > k + 1 − λ j and M 2 ,i,j = 0 for all i > k + 1 − µ j . By Lemma 4.7, we have for all 1 ≤ j ≤ k that M 1 ,i,j = 1 if i = w ( j ) and M 2 ,i,j = 1 if i = w c ( j ) , and M h,i,j = 0 , otherwise. Thus M respects P if and only if the indices of the nonzero entries M 1 ,w ( j ) ,j ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 17 and M 2 ,wc ( j ) ,j do not satisfy the inequalities w ( j ) > k + 1 − λ j or w c ( j ) > k + 1 − µ j , and this is precisely the claim. □ Using the pre vious fact we can no w show the number of hyperrook placements that respect P ⊆ ∆ corresponds to the conjectured number of nondegenerate hypermatrices that respect P at q = 1 . Theorem 4.9. F ix a format 2 × ( k + 1) × k and let P ⊆ ∆ be a plane partition. There ar e ( k + 1 − λ 1 ) · · · (2 − λ k ) · ( k − µ 1 ) · · · (2 − µ k − 1 )(1 − µ k ) elements ( w , c ) ∈ C k +1 such that w ( i ) ≤ k + 1 − λ i and w c ( i ) ≤ k + 1 − µ i for 1 ≤ i ≤ k . Remark 4.10. It’ s not hard to see that if P is a plane partition such that P ⊆ ∆ , then the given product is 0 . Pr oof. Let P ⊆ ∆ and w ∈ S k +1 with w ( i ) ≤ k + 1 − λ i for i ∈ [ k ] . W e will ﬁrst show the number of ( k + 1) -cycles c ∈ S k +1 such that w c ( i ) ≤ k + 1 − µ i for i ∈ [ k ] is ( k − µ 1 ) · ( k − 1 − µ 2 ) · · · (1 − µ k ) , independent of the choice of w . The general strategy we will use to prove this claim is to induct on k , so we need some w ay to relate these sets to a pre vious case. Since w ∈ S k +1 , there are k + 1 − µ i integers j ∈ [ k + 1] such that w ( j ) ≤ k + 1 − µ i . Also, since w ( i ) ≤ k + 1 − λ i and µ i ≤ λ i , we kno w w ( i ) ≤ k + 1 − µ i and particular w (1) ≤ k + 1 − µ 1 . For a ( k + 1) -c ycle c ∈ S k +1 we kno w c ( i )  = i , so if c is some ( k + 1) -cycle such that w c ( i ) ≤ k + 1 − µ i then c (1)  = 1 . If a 1 , a 2 , . . . , a k − µ 1 ∈ [ k + 1] are the k − µ 1 integers with a j  = 1 and w ( a j ) ≤ k + 1 − µ 1 , then we know c (1) = a j for some j = 1 , 2 , . . . , k − µ 1 , so there are ( k − µ 1 ) choices of c (1) . W e no w deﬁne the sets C 1 , C 2 , . . . , C k − µ 1 where C j := { c : c (1) = a j and w c ( i ) ≤ k + 1 − µ i for all i ∈ [ k ] } . W e will show that each of these sets has the same size, by giving bijections between them and a set coming from a plane partition in a smaller box. T o this end, we make some more deﬁnitions. Let P ′ = ⟨ λ ′ , µ ′ ⟩ be the plane partition of format 2 × k × ( k − 1) where λ ′ i = λ i +1 and µ ′ i = µ i +1 for i ∈ [ k − 1] . Since P ⊆ ∆ = ∆ 1 ,k we hav e λ i ≤ k + 1 − i and µ i ≤ k − i . Therefore, by deﬁnition λ ′ i ≤ k + 1 − ( i + 1) = k − i and µ ′ i ≤ k − ( i + 1) = ( k − 1) − i so P ′ ⊆ ∆ 1 ,k − 1 . W e now deﬁne w ′ ∈ S k by w ′ ( i ) = ( w ( i + 1) if w ( i + 1) < w (1) w ( i + 1) − 1 if w ( i + 1) > w (1) and we claim w ′ ( i ) ≤ k − λ ′ i for i ∈ [ k − 1] . W e have tw o cases: if w ′ ( i ) = w ( i + 1) then w ′ ( i ) < w (1) ≤ k + 1 − λ 1 ≤ k + 1 − λ i +1 = k + 1 − λ ′ i and if w ′ ( i ) = w ( i + 1) − 1 then w ′ ( i ) ≤ ( k + 1 − λ i +1 ) − 1 = k − λ i +1 = k − λ ′ i . Therefore w ′ ( i ) ≤ k − λ ′ i for all i ∈ [ k − 1] as claimed. W e now deﬁne B to be the set of k -cycles c ′ ∈ S k such that w ′ c ′ ( i ) ≤ k − µ ′ i for all i ∈ [ k − 1] W e construct a collection of bijections f 1 , f 2 , . . . , f k − µ 1 where f j : C j → B . For c ∈ C j with c = (1 c (1) · · · c k (1)) we deﬁne f j by f j ( c ) =  ( c (1) − 1) ( c 2 (1) − 1) · · · ( c k (1) − 1)  . This is clearly a k -cycle in S k , but for f j to be well deﬁned we must check that w ′ ( f j ( c )( i )) ≤ k − µ ′ i for i ∈ [ k − 1] . T o do this we consider two cases on i ∈ [ k − 1] . First, if i  = c k (1) − 1 , then by 18 B. KOPR OWSKI AND J. B. LEWIS deﬁnition f j ( c )( i ) = c ( i + 1) − 1 . Therefore w ′ ( f j ( c )( i )) = w ′ ( c ( i + 1) − 1) so we have the following two possibilities: w ′ ( f j ( c )( i )) = ( w ( c ( i + 1)) if w ( c ( i + 1)) < w (1) w ( c ( i + 1)) − 1 if w ( c ( i + 1)) > w (1) . If w ′ ( f j ( c )( i )) = w ( c ( i + 1)) then w ( c ( i + 1)) < w (1) ≤ k + 1 − µ 1 and therefore w ′ ( f j ( c )( i )) ≤ k − µ 1 ≤ k − µ i +1 = k − µ ′ i . If w ′ ( f j ( c )( i )) = w ( c ( i + 1)) − 1 then by the deﬁnition of C j we ha ve w ( c ( i + 1)) ≤ k + 1 − µ i +1 and so w ′ ( f j ( c )( i )) = w ( c ( i + 1)) − 1 ≤ k − µ i +1 = k − µ ′ i . Therefore if i  = c k (1) − 1 we conclude that w ′ ( f j ( c )( i )) ≤ k − µ ′ i . Now we prove the case when i = c k (1) − 1 . By deﬁnition f j ( c )( c k (1) − 1) = c (1) − 1 and again we have the two cases. First, if w ′ ( c (1) − 1) = w ( c (1)) < w (1) then since w (1) < k + 1 − µ 1 we hav e w ′ ( c (1) − 1) ≤ w (1) − 1 ≤ k − µ 1 . If w ′ ( c (1) − 1) = w ( c (1)) − 1 we have w ′ ( c (1) − 1) ≤ ( k + 1 − µ 1 ) − 1 = k − µ 1 , so either w ay so w ′ ( f j ( c )( c k (1) − 1)) ≤ k − µ 1 ≤ k − µ c k (1) = k − µ ′ c k (1) − 1 . Thus, w ′ ( f j ( c )( i )) ≤ k − µ ′ i for all i ∈ [ k − 1] so f j is well deﬁned for j = 1 , 2 , . . . , k − µ 1 . Next we sho w each f j is a bijection by constructing a family of in verse functions g j : B → C j for j = 1 , 2 , . . . , k − µ 1 . Let c ∗ ∈ B which we can write as c ∗ = (1 c ∗ (1) · · · c k − 1 ∗ (1)) , and since a j > 1 there exists some n > 0 such that c n ∗ (1) = a j − 1 . No w deﬁne g j ( c ∗ ) =  1 a j ( c n +1 ∗ (1) + 1) · · · ( c n − 1 ∗ (1) + 1)  , which is clearly a ( k + 1) -cycle in S k +1 with g j ( c ∗ )(1) = a j . But, to sho w that g j ∈ C j we still need to verify w ( g j ( c ∗ )( i )) ≤ k + 1 − µ i for i ∈ [ k ] . W e again consider two cases on i . If i  = c n − 1 ∗ (1) then by deﬁnition g j ( c ∗ )( i + 1) = c ∗ ( i ) + 1 . By the deﬁnition of w ′ we kno w w ( i + 1) ≤ w ′ ( i ) + 1 and thus w ( g j ( c ∗ )( i + 1)) = w ( c ∗ ( i ) + 1) ≤ w ′ c ∗ ( i ) + 1 ≤ k + 1 − µ ′ i = k + 1 − µ i +1 . Again by deﬁnition g j ( c ∗ )( c n − 1 ∗ (1) + 1) = 1 so w ( g j ( c ∗ )( c n − 1 ∗ (1) + 1)) = w (1) ≤ k + 1 − λ 1 ≤ k + 1 − µ c n − 1 ∗ (1)+1 . Finally , w ( g j ( c ∗ )(1)) = w ( a j ) and by assumption w ( a j ) ≤ k + 1 − µ 1 , as needed. Therefore w ( g j ( c ∗ )( i )) ≤ k + 1 − µ i for all i ∈ [ k ] so g j is well deﬁned for j = 1 , 2 , . . . , k − µ 1 . No w we sho w g j is a left and right in verse of f j . Let c ∈ C j with c = (1 a j c 2 (1) · · · c k (1)) , then g j ◦ f j ( c ) = g j  ( a j − 1) ( c 2 (1) − 1) · · · ( c k (1) − 1)  = (1 a j c 2 (1) · · · c k (1)) . For c ∗ ∈ B with c ∗ = (1 c ∗ (1) · · · c k − 1 ∗ (1)) we ha ve f j ◦ g j ( c ∗ ) = f j  1 a j ( c n +1 ∗ (1) + 1) · · · ( c n − 1 ∗ (1) + 1)  = f j  1 ( c n ∗ (1) + 1) ( c n +1 ∗ (1) + 1) · · · ( c n − 1 ∗ (1) + 1)  by assumption that c n ∗ ( i ) + 1 = a j . Then we can see f j  1 ( c n ∗ (1) + 1) ( c n +1 ∗ (1) + 1) · · · ( c n − 1 ∗ (1) + 1)  = ( c n ∗ (1) c n +1 ∗ (1) · · · c n − 1 ∗ (1)) = c ∗ ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 19 by the translation in variance of cycles. Therefore f j is a bijection which shows | C j | = | B | . Further- more, since any c such that w c ( i ) ≤ k + 1 − µ i is in exactly one set C j we conclude the number of such c is       k − µ 1 G j =1 C j       = k − µ 1 X j =1 | C j | = ( k − µ 1 ) | B | . Equipped with this pre vious fact we are ready to compute the number of ( k + 1) -cycles c ∈ S k +1 such that wc ( i ) ≤ k + 1 − λ i for some ﬁx ed w via induction. For 2 × ( n + 1) × n hypermatrices we induct on n . F or the base case n = 1 , any plane partition must be contained in ∆ 1 , 1 where λ 1 = 1 and µ 1 = 0 . For any w ∈ S 2 and any plane partition P we always have w c (1) ≤ 2 and w c (2) ≤ 2 , and so the single choice c = (1 2) holds for all w . So the number of such c is 1 − µ 1 = 1 as claimed. For n = k − 1 assume that for any plane partition P ⊆ ∆ 1 ,k − 1 and w ∈ S k with w ( i ) ≤ k − λ i for i ∈ [ k − 1] that the number of k -cycles c ∈ S k such that w c ( i ) ≤ k − µ i for i ∈ [ k − 1] , is (( k − 1) − µ 1 )(( k − 2) − µ 2 ) · · · (1 − µ k − 1 ) . No w let P ⊆ ∆ 1 ,k and w ∈ S k +1 with w ( i ) ≤ k + 1 − λ i for i ∈ [ k + 1] . By what we ha ve just sho wn the number of c such that wc ( i ) ≤ k + 1 − µ i is ( k − µ 1 ) | B | . By the inducti ve h ypothesis | B | = (( k − 1) − µ 2 )(( k − 2) − µ 3 ) · · · (1 − µ k ) , so the number of c is ( k − µ 1 )(( k − 1) − µ 2 )(( k − 2) − µ 3 ) · · · (1 − µ k ) , as claimed. No w we count the pairs ( w , c ) ∈ C k +1 that respect some P ⊆ ∆ . Let w ∈ S k +1 such that w ( i ) ≤ k + 1 − λ i for all i ∈ [ k + 1] . It is straightforward that the number of w is ( k + 1 − λ 1 )( k − λ 2 ) · · · (2 − λ k ) and for any w we hav e the same number of ( k + 1) -c ycles c such that w c ( i ) ≤ k + 1 − µ i for all i ∈ [ k + 1] . From our counting abov e we conclude the number of pairs ( w , c ) ∈ C k +1 that respect P is ( k + 1 − λ 1 )( k − λ 2 ) · · · (2 − λ k ) · ( k − µ 1 )(( k − 1) − µ 2 ) · · · (1 − µ k ) , as claimed. □ 4.2. Decomposing the space of nondegenerate hypermatrices. The rest of this section is dev oted to the proof of Theorem 4.1. The main idea of the proof is to combine the rook placements of Section 4.1 with the Bruhat decomposition of the general linear group. W e begin by establishing some deﬁnitions. Deﬁnition 4.11. For a permutation w ∈ S n , we say that a pair ( i, j ) of integers with 1 ≤ i < j ≤ n is an in v ersion of w if w ( i ) > w ( j ) , and we denote by In v( w ) the in version set In v( w ) = { ( i, j ) : ( i, j ) is an in v ersion of w } . For example, the permutation w = 25314 ∈ S 5 has Inv( w ) = { (1 , 4) , (2 , 3) , (2 , 4) , (2 , 5) , (3 , 4) } . Observe that, by deﬁnition, In v( w − 1 ) =  ( w ( j ) , w ( i )) : ( i, j ) ∈ In v( w )  , so in particular the two sets In v( w ) and Inv( w − 1 ) are equinumerous. Deﬁnition 4.12. For a permutation w ∈ S n , an n × n matrix A o ver a ﬁeld F is a NW w -augmented matrix if it has the follo wing properties: A w ( i ) ,i = 1 for all i ∈ [ n ] , if ( i, j ) is an in v ersion of w then the entry A w ( j ) ,i can be any element of F , and all other entries of A are equal to 0 . Similarly , a SE w -augmented matrix ov er a ﬁeld F is a matrix A with the following properties: A w ( i ) ,i = 1 for all i , if ( i, j ) is an in v ersion of w then the entry A w ( i ) ,j can be any element of F , and all other entries of A are equal to 0 . 20 B. KOPR OWSKI AND J. B. LEWIS W e deﬁne ∗ w to be the set of all NW w -augmented matrices and w ∗ to be the set of all SE w - augmented matrices. For e xample, if w = 25134 , then ∗ w =                  a b 1 1 c 1 d 1 1       : a, b, c, d ∈ F            and w ∗ =                  1 1 a 1 1 1 b c d       : a, b, c, d ∈ F            . In its most concrete form, the Bruhat decomposition is the follo wing statement. Theorem 4.13 (Bruhat decomposition [Sta12, §1.10]) . Let U r epr esent the gr oup of n × n in vertible upper triangular matrices over F . Then GL n ( F ) is equal to the disjoint union GL n ( F ) = G w ∈ S n U w ∗ , wher e furthermor e eac h matrix in U w ∗ has a unique r epresentation as a pr oduct B A with B ∈ U and A ∈ w ∗ . Remark 4.14. There are seven other equi v alent versions of Theorem 4.13: writing L for the group of n × n in v ertible lo wer triangular matrices and using the obvious notions and notations w ∗ and ∗ w for SW and NE augmented matrices, we can replace U w ∗ in the theorem with any of w ∗ L , ∗ w U , L ∗ w , w ∗ L , Lw ∗ , ∗ w U , or U ∗ w . One may show the equiv alence of these decompositions by v arious combinations of the operations of transpose and multiplication on the left or right by the antidiagonal permutation matrix. These eight different decompositions correspond to eight dif ferent orders in which one could do the relev ant version of Gaussian elimination: choosing piv ots from left to right and eliminating up, or choosing pi vots from top to bottom and eliminating left, and so on. Remark 4.15. In the case that F = F q is a ﬁnite ﬁeld, Theorem 4.13 provides a direct connection between the rook placements and in vertible matrices in Haglund’ s theorem [Hag98, Thm. 1] (Equa- tion (1) in the introduction): it’ s easy to see that for any A ∈ w ∗ and B ∈ U the matrix B A respects a partition λ if and only if w respects λ . Letting S λ be the set of permutations w ∈ S n that respect λ , it follo ws the number of in v ertible matrices in GL n ( F q ) that respect λ is X w ∈ S λ | U w ∗ | = X w ∈ S λ | U | · | w ∗ | = ( q − 1) n q ( n 2 ) X w ∈ S λ q | Inv( w ) | . Our approach to prov e Theorem 4.1 is to mimic the argument of Remark 4.15 in the higher- dimensional setting. T o wards this end, we make the follo wing deﬁnition. Deﬁnition 4.16. Giv en permutations σ ∈ S k +1 and π ∈ S k , a ( σ, π ) -augmented hypermatrix is a 2 × ( k + 1) × k hypermatrix of the form ( A, A ′ ) ◦ E where A ∈ σ ∗ and A ′ ∈ ∗ ( π − 1 ) . ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 21 By the deﬁnition of the action of GL k +1 × GL k , the formula in the preceding deﬁnition can be written e ven more concretely as ( A, A ′ ) ◦ E =        A        1 0 · · · 0 0 1 · · · 0 . . . . . . . . . . . . 0 0 · · · 1 0 0 · · · 0        ( A ′ ) T , A        0 0 · · · 0 1 0 · · · 0 0 1 · · · 0 . . . . . . . . . . . . 0 0 · · · 1        ( A ′ ) T        , where the matrix ( A ′ ) T being multiplied on the right belongs to ∗ π . The next three results extend the ﬁrst part of Remark 4.15 to our setting, sho wing how to reduce the problem of enumerating nondegenerate hypermatrices respecting a plane partition P to the problem of enumerating augmented hypermatrices respecting P . Proposition 4.17. Every nonde generate 2 × ( k + 1) × k hypermatrix over F q can be written in e xactly q − 1 ways as a composition of the form ( B , B ′ ) ◦ H wher e B ∈ U ⊂ GL k +1 ( F q ) and B ′ ∈ L ⊂ GL k ( F q ) ar e triangular and H is a ( σ, π ) -augmented hypermatrix for some permutations σ ∈ S k +1 , π ∈ S k . Pr oof. By Theorem 3.3, each nondegenerate hypermatrix of format 2 × ( k + 1) × k can be written in exactly q − 1 ways as a composition ( M , N ) ◦ E where ( M , N ) ∈ GL k +1 ( F q ) × GL k ( F q ) . By Theorem 4.13 and Remark 4.14, there is exactly one choice of permutations σ ∈ S k +1 and π ∈ S k and matrices A, A ′ , B , B ′ such that A ∈ σ ∗ , A ′ ∈ ∗ ( π − 1 ) , B ∈ GL k +1 ( F q ) is upper-triangular , and B ′ ∈ GL k ( F q ) is lo wer -triangular such that M = B A and N = B ′ A ′ . Since this is a group action, ( M , N ) ◦ E = ( B A, B ′ A ′ ) ◦ E =  ( B , B ′ ) · ( A, A ′ )  ◦ E = ( B , B ′ ) ◦  ( A, A ′ ) ◦ E  , with ( A, A ′ ) ◦ E the deﬁning form of a ( σ, π ) -augmented hypermatrix. Thus each pair ( M , N ) corresponds to exactly one composition of the desired kind. This completes the proof. □ Proposition 4.18. Let P be a plane partition inside the box of format 2 × ( k + 1) × k , let H be a hypermatrix over a ﬁeld F , and let B ∈ U ⊂ GL k +1 ( F ) and B ′ ∈ L ⊂ GL k ( F ) be triangular . Then H r espects P if and only if ( B , B ′ ) ◦ H does. Pr oof. When multiplying any matrix X by an upper-triangular matrix B on the left, the nonzero entries in the product B X all lie at positions that are nonzero in X , or directly abov e such positions (in the same column). Like wise, when multiplying any matrix X by an upper-triangular matrix ( B ′ ) T on the right, the nonzero entries in X ( B ′ ) T all lie at positions that are nonzero in X , or directly to the right of such positions (in the same ro w). It foll o ws immediately that if X respects the partition λ and B , ( B ′ ) T are upper-triangular , then B X ( B ′ ) T also respects λ . No w the result follo ws by considering X to be in turn the two faces of H . □ Corollary 4.19. Let P be a plane partition inside the box of format 2 × ( k + 1) × k . The number of nonde g enerate hypermatrices o ver F q that r espect P is q k 2 ( q − 1) 2 k X ( σ,π ) ∈ S k +1 × S k # { ( σ, π ) -augmented hypermatrices that r espect P } . Pr oof. By Proposition 4.17, each nondegenerate hypermatrix can be written in exactly q − 1 ways as ( B , B ′ ) ◦ H where H is a ( σ, π ) -augmented hypermatrix, for some σ ∈ S k +1 and π ∈ S k . Furthermore, by Proposition 4.18, the original hypermatrix respects P if and only if H does. The number of choices for B , B ′ is q ( k +1 2 ) ( q − 1) k +1 · q ( k 2 ) ( q − 1) k = q k 2 ( q − 1) 2 k +1 . Thus, there is a ( q − 1) -to- q k 2 ( q − 1) 2 k +1 correspondence between the set of nondegenerate hypermatrices that respect P and the collection (ov er all σ and π ) of ( σ, π ) -augmented hypermatrices that respect P . The result follo ws immediately . □ 22 B. KOPR OWSKI AND J. B. LEWIS The next result identiﬁes precisely which pairs ( σ , π ) contribute to the sum in Corollary 4.19. Proposition 4.20. F or any plane partition P inside the box of format 2 × ( k + 1) × k , ther e exist ( σ, π ) - augmented hypermatrices that r espect P if and only if the hyperr ook placement ( m σ , m π − 1 ) ◦ E r espects P . Pr oof. Since m σ ∈ σ ∗ and m π − 1 ∈ ∗ ( π − 1 ) , if the hyperrook placement ( m σ , m π − 1 ) ◦ E respects P then there is at least one ( σ, π ) -augmented hypermatrix that respects P . Con v ersely , suppose that there is a ( σ, π ) -augmented hypermatrix H = ( A, ( A ′ ) T ) ◦ E that re- spects P (so A ∈ σ ∗ and A ′ ∈ ∗ π ). Recall from Deﬁnition 4.4 the notations w = σ π and c = π − 1 (1 2 · · · k k + 1) π . By deﬁnition, we have for each j ∈ [ k ] that H 1 ,w ( j ) ,j = k X ℓ =1 A w ( j ) ,ℓ A ′ ℓ,j . By the deﬁnition of σ ∗ and w , we hav e that 1 = A w ( j ) ,σ − 1 w ( j ) = A w ( j ) ,π ( j ) = A w ( j ) ,π ( j ) is the unique 1 in row w ( j ) of A and that A w ( j ) ,ℓ = 0 for ℓ < π ( j ) . Similarly , by the deﬁnition of ∗ π , A ′ π ( j ) ,j = 1 is the unique 1 in column j of A ′ and A ′ ℓ,j = 0 for ℓ > π ( j ) . Thus H 1 ,w ( j ) ,j = 1 . Since H respects P , the position (1 , w ( j ) , j ) must not be contained in P . Similarly , we ha ve by deﬁnition that for each j ∈ [ k ] , H 2 ,wc ( j ) ,j = k +1 X ℓ =2 A wc ( j ) ,ℓ A ′ ℓ − 1 ,j . By the deﬁnition of σ ∗ , A wc ( j ) ,π c ( j ) = A wc ( j ) ,σ − 1 wc ( j ) = 1 and A wc ( j ) ,ℓ = 0 if ℓ < π c ( j ) . Similarly , by the deﬁnition of ∗ π , A ′ π ( j ) ,j = 1 and A ′ ℓ − 1 ,j = 0 if ℓ − 1 > π ( j ) . Furthermore, since j ≤ k , we hav e π c ( j ) = (1 2 · · · k + 1) π ( j ) = π ( j ) + 1 , and so the previous condition ℓ − 1 > π ( j ) can be re written as ℓ > π c ( j ) . Thus H 2 ,wc ( j ) ,j = 1 , and since H respects P , the position (2 , w c ( j ) , j ) must not be contained in P . Finally , since P does not contain any of the entries (1 , w ( j ) , j ) or (2 , w c ( j ) , j ) for 1 ≤ j ≤ k , it follo ws from Lemma 4.7 that the hyperrook placement ( m σ , m π − 1 ) ◦ E respects P . □ T o complete the proof of Theorem 4.1, it sufﬁces to show that the number of ( σ , π ) -augmented hypermatrices that respect P is a po wer of q when the hyperrook placement ( m σ , m π − 1 ) ◦ E respects P . In comparison with the argument in the matrix case (see Remark 4.15), this is signiﬁcantly complicated by the fact that a generic ( σ , π ) -augmented hypermatrix may not respect a plane partition P , e v en when the associated hyperrook placement does. Example 4.21. Consider the case with k = 4 , σ = 52134 , and π = 2314 . In this case, the generic element of σ ∗ is       0 0 1 0 0 0 1 x 2 , 3 0 0 0 0 0 1 0 0 0 0 0 1 1 x 1 , 2 x 1 , 3 x 1 , 4 x 1 , 5       , the generic element of ∗ π is     y 1 , 3 y 2 , 3 1 0 1 0 0 0 0 1 0 0 0 0 0 1     , ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 23 and so a generic ( σ, π ) -augmented hypermatrix is of the form             0 1 0 0 1 x 2 , 3 0 0 0 0 0 1 0 0 0 0 x 1 , 2 + y 1 , 3 x 1 , 3 + y 2 , 3 1 x 1 , 4       ,       1 0 0 0 x 2 , 3 + y 1 , 3 y 2 , 3 1 0 0 1 0 0 0 0 0 1 x 1 , 3 + x 1 , 2 y 1 , 3 x 1 , 4 + x 1 , 2 y 2 , 3 x 1 , 2 x 1 , 5             . Although the hyperrook placement ( m σ , m π − 1 ) ◦ E respects the plane partition P =  ⟨ 3 , 2 ⟩ , ⟨ 2 , 2 ⟩  , the generic ( σ, π ) -augmented hypermatrix does not. Example 4.21 sho ws that the enumeration of augmented h ypermatrices respecting a plane partition may be nontri vial. Example 4.22. Continue the choices of Example 4.21. It follo ws from the preceding computation that the number of ( σ, π ) -augmented h ypermatrices o ver F q that respect P is the number of solutions ov er F q to the system of algebraic equations          x 1 , 2 + y 1 , 3 = 0 x 1 , 3 + y 2 , 3 = 0 x 1 , 2 y 1 , 3 + x 1 , 3 = 0 x 1 , 2 y 2 , 3 + x 1 , 4 = 0 in the | In v( σ ) | + | In v( π ) | = 7 v ariables x 1 , 2 , . . . , y 2 , 3 . W e may solve this system as follows: ﬁrst, choose the values of x 2 , 3 , x 1 , 5 , and x 1 , 2 freely ( q 3 choices). The ﬁrst equation in the system determines the value of y 1 , 3 uniquely . Then the third equation determines the v alue of x 1 , 3 uniquely . Then the second equation determines the value of y 2 , 3 uniquely , and the last equation determines the v alue of x 1 , 4 uniquely . So in total there are q 3 ( σ, π ) -augmented hypermatrices o ver F q in this case. The ﬁnal stage in the proof of Theorem 4.1 is to sho w that the preceding argument can be carried out in general: that is, for each plane partition P and each hyperrook placement ( m σ , m π − 1 ) ◦ E that respects P , there exists a special subset of the free entries of σ ∗ and ∗ π such that for any choice of values for the elements of this subset, there is a unique ( σ, π ) -augmented hypermatrix with the special subset taking these v alues that respects P . In order to implement this approach, we use the follo wing labeling scheme for the free entries of an augmented matrix, which is already illustrated abov e in Example 4.21. Deﬁnition 4.23. Gi ven a permutation σ ∈ S k +1 , for each in v ersion ( i, j ) of σ , we write x i,j for the corresponding entry in position ( σ ( i ) , j ) of a generic matrix in σ ∗ . Like wise, given a permutation π ∈ S k , for each in v ersion ( i, j ) of π , we write y i,j for the entry in the corresponding position ( π ( j ) , i ) of a generic matrix in ∗ π . Our next result identiﬁes those positions in the front face of a ( σ, π ) -augmented hypermatrix that are not identically zero and lie belo w and to the left of the 1 s in the associated rook placement. These are precisely the entries that could pre vent a ( σ, π ) -augmented hypermatrix from respecting a plane partition P that is respected by the hyperrook placement ( m σ , m π − 1 ) ◦ E . The statement is illustrated in Figure 1(a). Proposition 4.24. F ix permutations ( σ , π ) ∈ S k +1 × S k and integ ers i, j such that 1 ≤ i < j ≤ k + 1 and w ( i ) < w ( j ) . Then (1 , w ( j ) , i ) is not identically zer o in the set of ( σ , π ) -augmented hypermatrices if and only if ( i, j ) is an in version of π (equivalently , j ≤ k and π ( j ) < π ( i ) ). Furthermor e if ( i, j ) is an in version of π , then the (1 , w ( j ) , i ) -entry is given by the polynomial x π ( j ) ,π ( i ) + y i,j + X t ∈ S i,j x π ( j ) ,π ( t ) y i,t 24 B. KOPR OWSKI AND J. B. LEWIS (a) i j w ( i ) w ( j ) 1 1 ∗ (b) i j w c ( i ) w c ( j ) 1 1 ∗ F I G U R E 1 . Left: the situation described in Proposition 4.24, with a potential λ in gray . Right: the situation described in Proposition 4.25, with a potential µ in gray . wher e the x - and y -variables ar e as in Deﬁnition 4.23 and S i,j = { t | ( i, t ) ∈ Inv( π ) and ( π ( j ) , π ( t )) ∈ Inv( σ ) } . Pr oof. Let A be a generic element of σ ∗ and let A ′ be a generic element of ∗ π . Therefore, by deﬁnition, the (1 , w ( j ) , i ) -entry of the ( σ, π ) -augmented hypermatrix ( A, ( A ′ ) T ) ◦ E is (2) k X ℓ =1 A w ( j ) ,ℓ A ′ ℓ,i . By the deﬁnition of σ ∗ and w , we hav e that 1 = A w ( j ) ,σ − 1 w ( j ) = A w ( j ) , π ( j ) is the unique 1 in ro w w ( j ) of A and that A w ( j ) ,ℓ = 0 for ℓ < π ( j ) . Similarly , A ′ π ( i ) ,i = 1 is the unique 1 in column i of A ′ and A ′ ℓ,i = 0 for ℓ > π ( i ) . It follows immediately that if π ( i ) < π ( j ) , then ev ery term in the sum (2) is equal to 0 , and consequently that the (1 , w ( j ) , i ) -entry of ( A, ( A ′ ) T ) ◦ E is identically equal to 0 in this case. No w suppose π ( i ) > π ( j ) . Then j < k + 1 and ( i, j ) is an inv ersion of π . Then the formula (2) for the (1 , w ( j ) , i ) -entry of ( A, ( A ′ ) T ) ◦ E can be re written as π ( i ) X ℓ = π ( j ) A w ( j ) ,ℓ A ′ ℓ,i = A w ( j ) ,π ( j ) A ′ π ( j ) ,i + π ( i ) − 1 X ℓ = π ( j )+1 A w ( j ) ,ℓ A ′ ℓ,i ! + A w ( j ) ,π ( i ) A ′ π ( i ) ,i = A ′ π ( j ) ,i + A w ( j ) ,π ( i ) + π ( i ) − 1 X ℓ = π ( j )+1 A w ( j ) ,ℓ A ′ ℓ,i . Since ( i, j ) is an in version of π , A ′ π ( j ) ,i = y i,j . Also, since w ( i ) < w ( j ) , we hav e σ ( π ( i )) = w ( i ) < w ( j ) = σ ( π ( j )) , so that ( π ( j ) , π ( i )) is an in v ersion of σ . Therefore A w ( j ) ,π ( i ) = x π ( j ) ,π ( i ) . Thus we can further simplify our expression for the (1 , w ( j ) , i ) -entry of ( A, ( A ′ ) T ) ◦ E to x π ( j ) ,π ( i ) + y i,j + π ( i ) − 1 X ℓ = π ( j )+1 A w ( j ) ,ℓ A ′ ℓ,i , and in particular we see that this entry is not identically zero. This completes the proof of the “if and only if ” part of the statement. T o ﬁnish, we need to identify the nonzero terms A w ( j ) ,ℓ A ′ ℓ,i that appear in the pre vious sum. By the deﬁnition of ∗ π , we kno w for ℓ < π ( i ) that the entry A ′ ℓ,i is generically nonzero if and only if ( i, π − 1 ( ℓ )) is an in v ersion of π , and so in particular that i < π − 1 ( ℓ ) . In this case, the entry is y i,π − 1 ( ℓ ) . Similarly , for ℓ > π ( j ) , the entry A w ( j ) ,ℓ = A σ ( π ( j )) ,ℓ is generically nonzero if and only if ( π ( j ) , ℓ ) is an in version of σ , or equiv alently if σ ( ℓ ) < w ( j ) , and in this case the entry is x π ( j ) ,ℓ . ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 25 Therefore, deﬁning t = π − 1 ( ℓ ) , the summand A w ( j ) ,ℓ A ′ ℓ,i in the last displayed equation above is nonzero exactly when ( i, t ) ∈ Inv( π ) and ( π ( j ) , π ( t )) ∈ Inv( σ ) , and in this case its v alue is x π ( j ) ,π ( t ) y i,t . This completes the proof. □ W e now do a similar analysis for the back face. The corresponding situation is illustrated in Figure 1(b). Proposition 4.25. F ix permutations ( σ, π ) ∈ S k +1 × S k and inte gers i, j such that 1 ≤ i < j ≤ k + 1 and w c ( i ) < w c ( j ) . Then the (2 , w c ( j ) , i ) -entry is not identically zer o in the set of ( σ, π ) -augmented hypermatrices if and only if ( i, j ) is an inver sion of π c . This is also equivalent to the condition that either ( i, j ) is an in version of π or j = k + 1 . Furthermore , if ( i, j ) is an inver sion of π then the (2 , w c ( j ) , i ) -entry is given by x π c ( j ) ,πc ( i ) + y i,j + X t ∈ S ′ i,j x π c ( j ) ,πc ( t ) y i,t , while if j = k + 1 then the (2 , w c ( j ) , i ) -entry is given by x π c ( j ) ,πc ( i ) + X t ∈ S ′ i,j x π c ( j ) ,πc ( t ) y i,t , wher e in both cases S ′ i,j = { t | ( i, t ) ∈ Inv( π ) and ( π c ( j ) , π c ( t )) ∈ Inv( σ ) } . Pr oof. Let A be a generic element of σ ∗ and let A ′ be a generic element of ∗ π . Therefore, by deﬁnition, the (2 , w c ( j ) , i ) -entry of the ( σ, π ) -augmented hypermatrix ( A, ( A ′ ) T ) ◦ E is (3) k +1 X ℓ =2 A wc ( j ) ,ℓ A ′ ℓ − 1 ,i . By the deﬁnition of σ ∗ , A wc ( j ) ,π c ( j ) = A wc ( j ) ,σ − 1 wc ( j ) = 1 and A wc ( j ) ,ℓ = 0 if ℓ < π c ( j ) . Similarly , by the deﬁnition of ∗ π , A ′ π ( i ) ,i = 1 and A ′ ℓ − 1 ,i = 0 if ℓ − 1 > π ( i ) . Furthermore, since i < k + 1 , we ha ve πc ( i ) = (1 2 · · · k + 1) π ( i ) = π ( i ) + 1 , and so the pre vious condition ℓ − 1 > π ( i ) can be re written as ℓ > πc ( i ) . It follows immediately that if π c ( i ) < πc ( j ) , then every term in the sum (3) is equal to 0 , and consequently that the (2 , w c ( j ) , i ) -entry of ( A, ( A ′ ) T ) ◦ E is identically equal to 0 in this case. No w suppose π c ( i ) > π c ( j ) , so ( i, j ) is an in v ersion of π c . Since wc ( i ) < w c ( j ) (by hypothesis) and w c = σ ( π c ) (by deﬁnition), this means that ( π c ( j ) , π c ( i )) is an in v ersion of σ , and A wc ( j ) ,π c ( i ) = x π c ( j ) ,πc ( i ) . Since A ′ ℓ − 1 ,i = 1 when ℓ = π ( i ) + 1 = π c ( i ) and A ′ ℓ − 1 ,i = 0 for ℓ > π ( i ) + 1 , the formula (3) for the (2 , w c ( j ) , i ) -entry of ( A, ( A ′ ) T ) ◦ E can be re written as x π c ( j ) ,πc ( i ) + π ( i ) X ℓ =2 A wc ( j ) ,ℓ A ′ ℓ − 1 ,i . In particular , we see that this entry is not identically zero in this case. This establishes the ﬁrst equi v alence, i.e., if ( i, j ) is an in version of π c then the (2 , wc ( j ) , i ) -entry is nonzero. For the second equi v alence, observe ﬁrst that π c ( k + 1) = 1 and so ( i, k + 1) ∈ In v( π c ) for all 1 ≤ i ≤ k . Furthermore, if j ≤ k then π c ( j ) = π ( j ) + 1 and π c ( i ) = π ( i ) + 1 , so ( i, j ) is an in v ersion of π c if and only if ( i, j ) is an in version of π . This establishes the second equiv alence. T o ﬁnish, we need to identify the indices ℓ ∈ [2 , π ( i )] that gi ve rise to nonzero terms A wc ( j ) ,ℓ A ′ ℓ − 1 ,i in the last displayed equation abov e. If j < k + 1 and ( i, j ) ∈ Inv( π c ) , then the rest of the proof is similar to the proof of Proposi- tion 4.24 after shifting e verything by c : deﬁne t = π − 1 ( ℓ − 1) and use the fact that π c ( t ) = 1 + π ( t ) 26 B. KOPR OWSKI AND J. B. LEWIS for 1 ≤ t ≤ k . So suppose instead that j = k + 1 . In that case, πc ( j ) = 1 and w c ( j ) = σ (1) , and the ℓ th term in the last displayed equation is nonzero exactly when both σ ( ℓ ) < σ (1) (in which case A wc ( j ) ,ℓ = A σ (1) ,ℓ = x 1 ,ℓ = x π c ( j ) ,ℓ ) and π − 1 ( ℓ − 1) > i (in which case A ′ ℓ − 1 ,i = y i,π − 1 ( ℓ − 1) = y i,c − 1 π − 1 ( ℓ ) ). Deﬁning t = c − 1 π − 1 ( ℓ ) = π − 1 ( ℓ − 1) , the conditions in the preceding sentence be- come w c ( t ) < w c ( j ) and t > i , while the restriction 2 ≤ ℓ ≤ π ( i ) becomes π c ( j ) < π c ( t ) < π c ( i ) . These conditions are equi v alent to the conditions that ( i, t ) ∈ In v( π ) = Inv( π c ) ∩ [ k ] 2 and ( π c ( t ) , π c ( j )) ∈ Inv( σ ) . This completes the proof. □ Next, we deﬁne a directed graph structure on the variables that appear in a generic ( σ, π ) -augmented hypermatrix. In the ﬁnal proof of Theorem 4.1, this structure will guide us in choosing a good order to solve for the v ariables to respect a gi v en plane partition, as in Example 4.22. Deﬁnition 4.26. Fix any π ∈ S k and σ ∈ S k +1 . Let Y π = { y i,j : ( i, j ) ∈ In v( π ) } be the set of v ariables in the generic element of ∗ π , and let X σ = { x i,j : ( i, j ) ∈ In v( σ ) } be the set of variables in the generic element of σ ∗ . Deﬁne a digraph D σ,π as follo ws: the nodes of D σ,π are the v ariables X σ ∪ Y π . T wo variables z 1 , z 2 are connected by an arc z 1 → z 2 if either of the following happen: there is some ( i, j ) ∈ Inv( π ) such that, in the (1 , w ( j ) , i ) -entry of the generic ( σ, π ) -augmented h ypermatrix gi ven by Proposition 4.24, z 2 = y i,j is the element of Y π appearing as a linear term and z 1 is any of the other variables that appear; or , there is some ( i, j ) ∈ Inv( π c ) such that, in the (2 , w c ( j ) , i ) -entry of the generic ( σ, π ) -augmented hypermatrix giv en by Proposition 4.25, z 2 = x π c ( j ) ,πc ( i ) is the element of X σ appearing as a linear term and z 1 is any of the other variables that appear . Example 4.27. The digraph D 52134 , 2314 corresponding to Example 4.21 contains the following arcs: • x 1 , 2 → y 1 , 3 , coming from entry (1 , 5 , 1) , with ( i, j ) = (1 , 3) , • x 1 , 3 → y 2 , 3 , coming from entry (1 , 5 , 2) , with ( i, j ) = (2 , 3) , • y 1 , 3 → x 2 , 3 , coming from entry (2 , 2 , 1) , with ( i, j ) = (1 , 3) , • x 1 , 2 → x 1 , 3 and y 1 , 3 → x 1 , 3 , coming from entry (2 , 5 , 1) , with ( i, j ) = (3 , 5) , and • x 1 , 2 → x 1 , 4 and y 2 , 3 → x 1 , 4 , coming from entry (2 , 5 , 2) , with ( i, j ) = (2 , 5) . By checking the deﬁnitions of S i,j and S ′ i,j in Propositions 4.24 and 4.25, we see that e very arc in D σ,π belongs to one of the follo wing six cases: for some ( i, j ) ∈ Inv( π ) such that w ( i ) < w ( j ) , it is of the form (a) x π ( j ) ,π ( i ) → y i,j , or there is a t ∈ [ k ] with π ( t ) < π ( i ) for which it is of the form (b) x π ( j ) ,π ( t ) → y i,j or (c) y i,t → y i,j with π ( j ) < π ( t ) , or for some ( i, j ) ∈ In v( π c ) such that wc ( i ) < w c ( j ) it is of the form (d) y i,j → x π c ( j ) ,πc ( i ) , or there is a t ∈ [ k ] with π ( t ) < π ( i ) for which it is of the form (e) y i,t → x π c ( j ) ,πc ( i ) or (f) x π c ( j ) ,πc ( t ) → x π c ( j ) ,πc ( i ) . Our ne xt result will be used in the ﬁnal stage of the proof to produce a linear order on the v ariables that we can use to solve for them one-by-one, as in Example 4.22. Proposition 4.28. F or any permutations σ ∈ S k +1 , π ∈ S k , the digraph D σ,π is acyclic. ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 27 Pr oof. Consider any (directed) walk W in D σ,π . W e seek to sho w that W cannot be a cycle. If W visits only vertices in X σ , then the arcs it follows all belong to case (f). Each such arc is of the form x π c ( j ) ,πc ( t ) → x π c ( j ) ,πc ( i ) with π c ( t ) = π ( t ) + 1 < π ( i ) + 1 = π c ( i ) ; since the second coordinate strictly increases at each step, no such walk can form a c ycle. Similarly , if W visits only vertices in Y π , then the arcs it follows all belong to case (c). Each such arc is of the form y i,t → y i,j with π ( j ) < π ( t ) ; since the value of π applied to the second index strictly decreases at each step, no such walk can form a c ycle. Otherwise, W passes through vertices in both X σ and Y π . W ithout loss of generality , we may assume that W starts and ends in Y π , so for some m ≥ 1 we can write W as W 1 → W ′ 1 → W 2 → W ′ 2 → · · · → W m → W ′ m → W m +1 where W 1 , W 2 , . . . , W m +1 are walks (possibly of length 0 ) on the subgraph of D σ,π induced by Y π and W ′ 1 , W ′ 2 , . . . , W ′ m are walks on the subgraph induced by X σ . W e deﬁne a weight function wt : X σ ∪ Y π → Z > 0 by wt( y i,j ) = π ( i ) and wt( x i,j ) = j. By checking each of the six cases into which arcs of D σ,π can fall, we see that if z 1 → z 2 in D σ,π then wt( z 1 ) ≤ wt( z 2 ) , and moreov er that if z 1 ∈ Y π and z 2 ∈ X σ then actually wt( z 1 ) < wt( z 2 ) (speciﬁcally parts (d) and (e) of the deﬁnition, since π c ( i ) = π ( i ) + 1 for i ∈ [ k ] ). Since W contains at least one edge from a y -v ariable to an x -v ariable, it also cannot be a cycle. This completes the proof. □ Theorem 4.29. F or any permutations ( σ , π ) ∈ S k +1 × S k and any plane partition P of format 2 × ( k + 1) × k , the number of ( σ , π ) -augmented hypermatrices o ver F q that r espect P is either 0 or a power of q , wher e in the latter case the e xponent depends on σ , π , and P but is independent of q . Pr oof. Fix P and ( σ, π ) . If the hyperrook placement ( m σ , m π − 1 ) ◦ E does not respect P , then by Proposition 4.20 there are no ( σ , π ) -augmented hypermatrices that respect P . So suppose instead that ( m σ , m π − 1 ) ◦ E respects P . By Proposition 4.28, the digraph D σ,π is acyclic; therefore it induces a partial order on the set of v ariables. Choose any linear e xtension of this partial order . This is a total ordering of the v ariables in X σ ∪ Y π such that, if z 1 and z 2 are two of these variables such that z 1 → z 2 (as in Deﬁnition 4.26), then z 1 will precede z 2 in this total order . Let A be a generic ( σ , π ) -augmented hypermatrix, and consider an entry A a 1 ,a 2 ,a 3 of A that is not identically zero and that lies inside P . The total order in the pre vious paragraph induces a total order on the v ariables that appear in A a 1 ,a 2 ,a 3 . By the construction of D σ,π , the last v ariable in A a 1 ,a 2 ,a 3 with respect to the total order is a linear term. Furthermore, it is easy to see from Deﬁnition 4.26 and Propositions 4.24 and 4.25 that the last variables for different entries of A are distinct. Therefore, if we count the assignments of the variables in X σ ∪ Y π to elements of F q with the property that the resulting ( σ, π ) -augmented hypermatrix respects P , assigning values to variables in the order gi v en by the total order , we ha ve q choices for each v ariable that is not the last in its entry (i.e., no restriction on the choice) and 1 choice for each variable that is the last in its entry , if that entry lies in P (namely , the (unique) v alue that makes the (unique) entry in which it is the last variable equal to 0 , gi v en the pre vious choices). Since the number of choices for each assignment is independent of the choices made, the total number of v alid assignments is the product of the number of choices for each variable. This is always a po wer of q (with exponent depending only on σ, π , P ), as claimed. □ W e are now ready to pro ve Theorem 4.1, which we restate for con venience. Theorem 4.1. Let P ⊆ ∆ . Ther e is a polynomial f with nonnegative inte ger coefﬁcients suc h that f (1) = ( k + 1 − λ 1 )( k − λ 2 ) · · · (2 − λ k ) · ( k − µ 1 )(( k − 1) − µ 2 ) · · · (1 − µ k ) 28 B. KOPR OWSKI AND J. B. LEWIS is the number of hyperr ook placements that r espect P and, for every prime power q , the number of nonde g enerate hypermatrices that r espect P is q k 2 ( q − 1) 2 k · f ( q ) . Pr oof. Deﬁne for each prime po wer q and each plane partition P of format 2 × ( k + 1) × k the function (4) e f ( q ) = X ( σ,π ) ∈ S k +1 × S k # { ( σ, π ) -augmented hypermatrices o ver F q that respect P } . By Corollary 4.19, the number of nondegenerate hypermatrices over F q that respect the plane partition P is q k 2 ( q − 1) 2 k · e f ( q ) . By Proposition 4.20, the ( σ, π ) -summand in (4) is nonzero exactly when ( m σ , m π − 1 ) ◦ E respects P ; and in that case, by Theorem 4.29, the summand is a power of q (whose exponent depends on σ , π , and P , but not on q ). It follows that there is a polynomial f ( x ) ∈ Z ≥ 0 [ x ] such that e f ( q ) = f ( q ) for all prime po wers q , and furthermore that f (1) is equal to the number of pairs ( σ, π ) ∈ S k +1 × S k such that the hyperrook placement ( m σ , m π − 1 ) ◦ E respects P . Then the result follo ws directly from Theorem 4.9. □ 5. F U RT H E R R E S E A R C H W e end with some open questions and some ideas for future research. 5.1. Pro ving the main conjecture. The main problem raised by our work is to pro ve Conjecture 3.1 in full generality . One approach would be to build on our proof of Theorem 4.1, which establishes that the number of nondegenerate hypermatrices o v er F q that respect the plane partition P is q k 2 ( q − 1) 2 k X ( σ,π ) ∈ S k +1 × S k : ( m σ ,m π − 1 ) ◦ E respects P q | Inv( σ ) | + | Inv( π ) |− h ( σ ,π ; P ) where h ( σ, π ; P ) is deﬁned to be the number of entries in a generic ( σ , π ) -augmented hypermatrix that are not identically zero and lie inside P ; these are characterized by Propositions 4.24 and 4.25. Thus, the main conjecture could be proved with “only combinatorics”, if one could understand the quantity h ( σ , π ; P ) well enough to ﬁnd a way to group terms correctly in order to giv e the desired factored form. Our attempts to carry this out have thus f ar been unsuccessful. 5.2. Other f ormats. T o what extent do our results extend to other formats? Our work relies in an essential way on Lemma 2.5 and Theorem 3.3 of Aitken, which hold respectively only for three- dimensional boundary formats and the speciﬁc format 2 × ( k + 1) × k . Is there an analogous version of Lemma 2.5 for higher-dimensional hypermatrices? If so, that might allo w one to extend Theorem 2.18 (which identiﬁes the unique maximal shape that allo ws nondegenerate hypermatrices for all three- dimensional boundary formats) to higher dimensions. A simple dimension count shows that we should not expect Theorem 3.3 to extend to any larger formats; 2 could it nonetheless still be the case that the nonde generate hypermatrices respecting a plane partition enumerate nicely? It seems challenging to test this question computationally . 2 For example, the space of 3 × 3 × 5 hypermatrices is 45 -dimensional, whereas the group GL 3 × GL 3 × GL 5 is only 9 + 9 + 25 = 43 -dimensional; the space of nondegenerate hypermatrices is presumably full-dimensional, so we should expect a two-parameter f amily of orbits of nondegenerate hypermatrices of this format under the lar ger group. ENUMERA TION OF NONDEGENERA TE 2 × ( k + 1) × k HYPERMA TRICES 29 5.3. Intrinsic deﬁnition of h yperr ook placements. Full-rank rook placements on a board can be described as arrangements ha ving one rook in each ro w and column; they can also be described as the result of multiplying the identity matrix by a permutation matrix. Our Deﬁnition 4.2 of hyperrook placements is analogous to the second deﬁnition. Do the same objects have a natural description along the lines of the ﬁrst deﬁnition? It follows immediately from Deﬁnition 4.2 that every hyperrook placement has a single 1 in each column of each face and at most 1 in each row of each face. One is tempted to guess that the hyper- rook placements are simply pairs ( S 1 , S 2 ) where S 1 , S 2 are placements of k rooks on the rectangle R ( k +1) × k with empty intersection and not sharing the same empty ro w . Howe ver , it’ s easy to see that the number of such pairs is strictly larger than the number of h yperrook placements when k > 2 . There are v arious deﬁnitions of three-dimensional rook placement in the literature that do not agree with ours (e.g., Latin squares, or the notion in [Zin07, AK13, Sch25]); typically , these are on cubical boards. 5.4. Lower rank. In the case of square matrices, the nondegenerac y condition (that the determinant is nonzero) can be equiv alently characterized as a rank condition (that the matrix has full rank), and the notion of a rook placements, q -rook numbers, etc., e xtends to lo wer ranks in a natural way , placing fe wer than the maximum number of rooks and thinking of rook placements as partial permutations. One might ask whether there is similarly a lo wer-rank v ersion of our theory . There are numerous inequi v alent deﬁnitions of the rank of a hypermatrix (see, e.g., [JN23] for a clear description of four such notions). Although these ha ve sometimes been discussed in the context of the hyperdetermi- nant (e.g., [Ott13, A Y23]), as far as we know , none of them has a tight connection to our notion of nondegenerac y . Moreover , basic questions about tensor rank (computing the rank of a hypermatrix, determining the orbits under the GL( F ) × · · · × GL( F ) action, or counting hypermatrices by rank ov er a ﬁnite ﬁeld) are all difﬁcult, see, e.g., [LS15, LS17, AL20, SL21, SZH25]. It is unclear whether a nice lo wer-rank theory can be based on an y of these ideas. Alternati vely , one might hope to construct a lo wer -rank hyperrook theory along the following lines (mentioned brieﬂy in the introduction): for a partition-shaped board B in the n × n square, its rook numbers r i ( B ) (the number of ways of placing i rooks on B ) and hit numbers h i ( B ) (the number of ways of placing n rooks on the square such that i of them lie in B ) are related by (5) X i r i ( B ) · ( x − 1) i · ( n − i )! = X i h i ( B ) · x i (and a q -analogue of this identity holds for q -rook and q -hit numbers). The deﬁnition of hit number extends immediately to our setting, by considering hyperrook placements on the 2 × ( k + 1) × k box. If we write do wn the generating function for these numbers, is there a natural change of basis (analogous to (5)) so that the coef ﬁcients can be interpreted as lo wer-rank rook numbers? A C K N O W L E D G E M E N T S This paper is based in part on the ﬁrst author’ s undergraduate honors thesis [K op23]. The ﬁrst author thanks the George W ashington Uni versity Enosinian Scholars Program and its participants for helpful discussions. W e thank Joe Bonin and Sam F . Hopkins for their comments on the thesis version. W ork of the second author was supported in part by a gift from the Simons Foundation (MPS-TSM-00006960). R E F E R E N C E S [Ait19] C. Aitken. Nondegenerate 2 × k × ( k + 1) hypermatrices. Linear and Multilinear Algebr a , 67:697 – 704, 2019. [AK13] F . Alayont and N. Krzywonos. Rook polynomials in three and higher dimensions. 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Remmel. q -counting rook conﬁgurations and a formula of Frobenius. J ournal of Combi- natorial Theory , Series A , 41:246–275, 1986. [Hag98] J. Haglund. q -rook polynomials and matrices over ﬁnite ﬁelds. Advances in Applied Mathematics , 20:450–487, 1998. [JN23] M. Juvekar and A. Nadjimzadah. Notions of tensor rank. Online J. Anal. Comb . , (18):Paper No. 5, 2023. [K op23] B. Kopro wski. Enumeration of nondegenerate 2 × k × ( k + 1) hypermatrices. Undergraduate honors thesis, George W ashington University , 2023. [KR46] I. Kaplansk y and J. Riordan. The problem of the rooks and its applications. Duke Mathematical Journal , 13:259– 268, 1946. [LS15] M. La vrauw and J. Sheeke y . Canonical forms of 2 × 3 × 3 tensors ov er the real ﬁeld, algebraically closed ﬁelds, and ﬁnite ﬁelds. Linear Algebra Appl. , 476:133–147, 2015. [LS17] M. Lavrauw and J. Sheek ey . Classiﬁcation of subspaces in F 2 ⊗ F 3 and orbits in F 2 ⊗ F 3 ⊗ F r . J. Geom. , 108(1):5–23, 2017. [MY08] G. Musiker and J. Y u. The 2 × 2 × 2 hyper-determinant and its enumeration over F q . Unpublished manuscript, 2008. [Ott13] G. Ottaviani. Introduction to the hyperdeterminant and to the rank of multidimensional matrices. In Commutative algebra , pages 609–638. Springer , New Y ork, 2013. [Sch25] J. Schabanel. 3D permutations and triangle solitaire. , 2025. [SL21] S.G. Stavrou and R.M. Low . The maximum rank of 2 × · · · × 2 tensors over F 2 . Linear Multilinear Algebr a , 69(3):394–402, 2021. [Sta12] R. Stanley . Enumerative Combinatorics , v olume 1. Cambridge Univ ersity Press, Ne w Y ork, NY , 2nd edition, 2012. [SZH25] X. Song, B. Zheng, and R. Huang. On the rank of m × 2 × 2 and m × 3 × 2 tensors over arbitrary ﬁelds. Electr onic J ournal of Linear Algebr a , 41:353–366, 2025. [Zin07] B. Zindle. Rook polynomials for chessboards of two and three dimensions. Master’ s thesis, Rochester Institute of T echnology , 2007. D E PART M E N T O F M A T H E M A T I C S , N O RT H C A RO L I N A S TA T E U N I V E R S I T Y , R A L E I G H , N C , U S A Email addr ess : bmkoprow@ncsu.edu D E PART M E N T O F M A T H E M A T I C S , G E O R G E W A S H I N G T O N U N I V E R S I T Y , W A S H I N G T O N , D C , U S A Email addr ess : jblewis@gwu.edu

Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

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