Counting 3-way contingency tables via quiver semi-invariants

COUNTING 3-W A Y CONTINGENCY T ABLES VIA QUIVER SEMI-INV ARIANTS CALIN CHINDRIS AND DEEP ANSHU PRASAD A B S T R A C T . Let T a , b be the number of 3 -way contingency tables of size m × n × p with two of its three plane-sum mar gins fixed by a = ( a 1 , . . . , a m ) ∈ N m and b = ( b 1 , . . . , b n ) ∈ N n . When p = 1 , this is the number of m × n non-negative integer matrices whose row and column sums are fixed by a and b . In this paper , we study the numbers T a , b through the lens of quiver invariant theory . Let Q p m,n be the p -complete bipartite quiver with m source vertices, n sink vertices, and p arrows from each sour ce to each sink. Let 1 denote the dimension vector of Q p m,n that takes value 1 at every vertex of Q p m,n , and let θ a , b denote the integral weight that assigns a i to the i th source vertex and − b j to the j th sink vertex of Q p m,n . W e begin by realizing T a , b as the dimension of the space of semi-invariants associated to ( Q p m,n , 1 , θ a , b ) . Using this connection and methods from quiver invariant theory , we show that T a , b is a parabolic Kostka coef ficient. In the case p = 1 , this recovers the formula for the number of the m × n contingency tables with row and column sums fixed by a and b , which in the classical 2 -way setting can also be obtained via the Robinson-Schensted-Knuth correspondence. C O N T E N T S 1. Introduction 1 2. W eight spaces of quiver semi-invariants: the tools 2 3. The number of 3 -way contingency tables as parabolic Kostka coefficients 6 References 10 1. I N T R O D U C T I O N Let m , n , and p be positive integers, and let a = ( a 1 , . . . , a m ) and b = ( b 1 , . . . , b n ) be two vectors with non-negative integer coefficients such that N := m X i =1 a i = n X j =1 b j . Date : March 27, 2026. 2010 Mathematics Subject Classification. 16G20, 13A50, 14L24. Key words and phrases. Contingency tables, Littlewood-Richardson coefficients, quiver exceptional se- quences, quiver semi-invariants. 1 Let T a , b be the number of all 3 -way contingency tables X = ( x ij k ) ( i,j,k ) ∈ [ m ] × [ n ] × [ p ] whose entries x i,j,k are non-negative integers such that (1) n X j =1 p X k =1 x ij k = a i , ∀ i ∈ [ m ] , and (2) m X i =1 p X k =1 x ij k = b j , ∀ j ∈ [ n ] . In what follows, K λ, R denotes the parabolic Kostka coefficient associated to a partition λ and a sequence R = { R 1 , . . . , R s } of rectangular partitions. Thus K λ, R is the multiplicity of the irr educible r epresentation of GL( r ) of highest weight λ in the tensor pr oduct of the irreducible representations with highest weights R 1 , . . . , R s . W e assume that λ has at most r non-zer o parts and that each R i has height at most r . Theorem 1.1. W ith the notation as above, (3) T a,b = K λ, R , where λ = (( pN ) pm ) and R is the following sequence of rectangular partitions (4) R = { ( N ( p − 1) m ) , . . . , ( N ( p − 1) m ) | {z } p times , ( a pm − 1 1 ) , . . . , ( a pm − 1 m ) , ( b 1 ) , . . . , ( b n ) } . T o prove our formula, we first expr ess T a , b as the dimension of a weight space of semi- invariants for a p -complete bipartite quiver . W e then apply general r eduction techniques, such as the Embedding Theorem 2.6 for quiver semi-invariants, to r educe the problem to computing semi-invariants for a star quiver . These, in turn, can be expressed as parabolic Kostka numbers. W e note that when p = 1 , T a , b is the number of m × n non-negative integer matrices whose r ow and column sums are fixed by a and b . In this case, ( 3 ) r ecovers the classical formula arising from the Robinson–Schensted–Knuth correspondence, but by a quiver- invariant-theoretic argument that does not use the correspondence itself; see Cor ollary 3.2 for full details. 2. W E I G H T S P A C E S O F Q U I V E R S E M I - I N V A R I A N T S : T H E T O O L S Throughout, we work over the field C of complex numbers and denote by N = { 0 , 1 , . . . } . For a positive integer L , we denote by [ L ] = { 1 , . . . , L } . A quiver Q = ( Q 0 , Q 1 , t, h ) consists of two finite sets Q 0 (vertices) and Q 1 (arrows) together with two maps t : Q 1 → Q 0 (tail) and h : Q 1 → Q 0 (head). W e repr esent Q as a directed graph with set of vertices Q 0 and directed edges a : ta → ha for every a ∈ Q 1 . A r epr esentation of Q is a family V = ( V x , V a ) x ∈Q 0 ,a ∈Q 1 , wher e V x is a finite-dimensional C -vector space for every x ∈ Q 0 , and V a : V ta → V ha is a C -linear map for every a ∈ Q 1 . After fixing bases for the vector spaces V x , x ∈ Q 0 , we often think of the linear maps V a , a ∈ Q 1 , as matrices of appropriate size. A morphism φ : V → W between two r epr esentations is a collection ( φ x )) x ∈Q 0 of C -linear maps with φ x ∈ Hom C ( V x , W x ) for every x ∈ Q 0 , and such that φ ha ◦ V a = W a ◦ φ ta for every a ∈ Q 1 . The C -vector space of all morphisms from V to W is denoted by Hom Q ( V , W ) . 2 The dimension vector dim V ∈ N Q 0 of a repr esentation V is defined by ( dim V ) x := dim C V x for all x ∈ Q 0 . By a dimension vector of Q , we simply mean an N -valued function on the set of vertices Q 0 . W e say a dimension vector β is sincere if β x > 0 for every x ∈ Q 0 . The simple dimension vector at x ∈ Q 0 , denoted by e x , is defined by e x ( y ) = δ x,y , ∀ y ∈ Q 0 , where δ x,y is the Kronecker symbol. The Euler form (also known as the Ringel form) of Q is the bilinear form on Z Q 0 defined by ⟨ α, β ⟩ := X x ∈Q 0 α x β x − X a ∈Q 1 α ta β ha , ∀ α , β ∈ Z Q 0 . From now on, we assume that all of our quivers are finite, connected, and acyclic. Then, for any integral weight σ ∈ Z Q 0 , there exists a unique α ∈ Z Q 0 such that σ x = ⟨ α, e x ⟩ , ∀ x ∈ Q 0 . Let β be a sincere dimension vector of a quiver Q , and let us consider the repr esentation space of β -dimensional repr esentations of Q , rep( Q , β ) := Y a ∈Q 1 C β ha × β ta . The base change gr oup GL( β ) := Q x ∈Q 0 GL( β x ) acts on rep( Q , β ) by simultaneous con- jugation, i.e. , for g = ( g x ) x ∈Q 0 ∈ GL( β ) and W = ( W a ) a ∈Q 1 ∈ rep( Q , β ) , we define g · W ∈ rep( Q , β ) by ( g · W ) a := g ha · W a · g − 1 ta , ∀ a ∈ Q 1 . This action descends to that of the subgroup SL( β ) := Y x ∈Q 0 SL( β x ) , giving rise to a highly non-trivial ring of semi-invariants SI( Q, β ) := C [rep( Q , β )] SL( β ) . (W e point out that since Q is assumed to be acyclic, the invariant ring C [rep( Q , β )] GL( β ) is precisely C .) Since GL( β ) is linearly reductive and SL( β ) is its commutator subgroup, we have the weight space decomposition SI( Q , β ) = M χ ∈ X ∗ (GL( β )) SI( Q , β ) χ , where X ∗ (GL( β )) is the group of rational characters of GL( β ) and SI( Q , β ) χ := { f ∈ C [rep( Q , β )] | g · f = χ ( g ) f , ∀ g ∈ GL( β ) } is the space of semi-invariants of weight χ . Every integral weight σ ∈ Z Q 0 defines a character χ σ of GL( β ) by (5) χ σ ( g ) := Y x ∈Q 0 (det g x ) σ x for all g = ( g x ) x ∈Q 0 ∈ GL( β ) . Moreover , every character of GL( β ) is of the form χ σ for some integral weight σ ∈ Z Q 0 . If β is sincer e, this gives a one-to-one correspondence, so we may identify the character group with Z Q 0 . In what follows, we write SI( Q , β ) σ for SI( Q , β ) χ σ . W e state a r eduction tool that will come on handy when pr oving Theorem 1.1 . It allows us to remove a vertex of weight zer o, provided its dimension is at least that of the head of the unique outgoing arrow . 3 Lemma 2.1 ( Removing vertices of zero weight ; see Lemma 4.6 in [ Chi09 ]) . Let Q be a quiver and v 0 a vertex such that near v 0 , Q looks like: v 0 w v 1 v ℓ . . . a 1 a ℓ b Suppose that β is a dimension vector and σ is a weight such that β ( v 0 ) ≥ β ( w ) and σ ( v 0 ) = 0 . Let Q be the quiver defined by Q 0 = Q 0 \ { v 0 } and Q 1 =  Q 1 \ { b, a 1 , . . . , a ℓ }  ∪ { ba 1 , . . . , ba ℓ } . If β = β | Q and σ = σ | Q are the r estrictions of β and σ to Q then SI( Q , β ) σ ∼ = SI( Q , β ) σ . Remark 2.2. W e note that, although the proof in [ Chi09 ] is rather short, its main ingr edient is the Fundamental Theorem for GL( n ) . 2.1. Network flow polytopes. When the dimension vector is equal to one at every vertex of Q , one can describe the dimensions of the corresponding spaces of semi-invariants in terms of network flows. This description that allows us to express T a , b as dimensions of weight spaces of quiver semi-invariants. Let σ ∈ Z Q 0 be an integral weight of Q . The network flow polytope associated to ( Q , σ ) is defined by (6) P σ :=      ( x a ) a ∈Q 1 ∈ R Q 1 ≥ 0        X a ∈Q 1 ta = x x a − X a ∈Q 1 ha = x x a = σ x for all x ∈ Q 0      . Lemma 2.3 ( Network flows from quiver semi-invariants ; see Lemma in [ Chi09 ]) . Let 1 be the dimension vector that is equal to one at every vertex of Q , and let σ ∈ Z Q 0 be a weight. Then dim C SI( Q , 1 ) σ =   P σ ∩ Z Q 0   . 2.2. Reflection transformations. W e will also make use of reflection transformations to establish ( 3 ) . Let β ∈ Z Q 0 ≥ 0 be a dimension vector , σ ∈ Z Q 0 , an integral weight and a x ∈ Q 0 a vertex. W e define s x Q to be the quiver obtained from Q by r eversing all arrows that start or end in x . W e also define s x β ∈ Z Q 0 by ( s x β ) y =      β y , if x  = y , − β x + X edges x — z β z , if x = y , and s x σ ∈ Z Q 0 by ( s x σ ) y = ( − σ x , if y = x, σ y + σ x b xy , if x  = y , 4 where b xy is the number of edges between x and y . Theorem 2.4. Suppose that x is a sink or a source vertex and s x β is a dimension vector , i.e. , s x β ∈ Z Q 0 ≥ 0 . Then (7) SI( Q , β ) σ ≃ SI( s x Q , s x β ) s x σ . 2.3. Quiver exceptional sequences. The notion of a quiver exceptional sequence, which we review below , plays a key role in our computations, since it allows us to reduce the problem to computing the dimensions of weight spaces of semi-invariants for a star quiver . These can then be expressed as parabolic Kostka coef ficients. In what follows, by a Schur repr esentation V of Q , we mean a repr esentation such that dim End Q ( V ) = 1 , that is, End Q ( V ) = { ( λ Id V ( x ) ) x ∈Q 0 | λ ∈ C } . For two dimension vectors α and β , we define ( α ◦ β ) Q := dim SI( Q , β ) ⟨ α, ·⟩ . (W e drop the subscript Q whenever Q is understood from the context.) It follows fr om the main results in [ DW00 ] that α ◦ β  = 0 if and only if ⟨ α , β ⟩ = 0 and Hom Q ( V , W ) = 0 for some repr esentations V and W of dimension vectors α and β , respectively . Definition 2.5 ( Quiver Exceptional Sequences ) . A sequence E = ( ε 1 , . . . , ε N ) of dimension vectors of Q is said to be a quiver exceptional sequence if: (1) ⟨ ε i , ε i ⟩ = 1 and ε i is the dimension vector of a Schur repr esentation for all i ∈ [ N ] ; (2) ⟨ ε i , ε j ⟩ ≤ 0 and ε j ◦ ε i  = 0 for all 1 ≤ i < j ≤ N . T o any quiver exceptional sequence E = ( ε 1 , . . . , ε N ) , we associate the quiver Q ( E ) with vertices { 1 , . . . , N } and −⟨ ε i , ε j ⟩ arrows fr om vertex i to vertex j for all 1 ≤ i  = j ≤ N . Let (8) I : R N → R Q 0 be the map defined by I ( γ 1 , . . . , γ N ) := N X i =1 γ i ε i for all γ = ( γ 1 , . . . , γ N ) ∈ R N . W e are now ready to state Derksen–W eyman’s Embedding Theorem. Theorem 2.6 ( The Embedding Theorem for Quiver Semi-Invariants ; see [ DW11 ]) . Let E = ( ε 1 , . . . , ε N ) be a quiver exceptional sequence for Q . If α and β ar e two dimension vectors of Q ( E ) , then ( α ◦ β ) Q ( E ) = ( I ( α ) ◦ I ( β )) Q . 5 3. T H E N U M B E R O F 3 - WA Y C O N T I N G E N C Y TA B L E S A S PA R A B O L I C K O S T K A C O E FFI C I E N T S In this section, we will work with the p -complete bipartite quiver with set of source vertices { x 1 . . . , x m } , set of sink vertices { y 1 , . . . y n } , and such that there ar e p arr ows from any source vertex x i to any sink vertex y j : x 1 . . . x m y 1 . . . y n Q p m,n : . . . p arrows . . . . . . . . . Recall that θ a , b is the integral weight that assigns a i to the i th source vertex and − b j to the j th sink vertex of Q p m,n . Then the network flow polytope P θ a , b is the set of all 3 -way contingency tables with mar gins given by ( 1 ) and ( 2 ) . It now follows fr om Lemma 2.3 that (9) T a , b = dim SI( Q p m,n , 1 ) θ a , b . Next, we explain how to simplify the task of computing semi-invariants for Q p m,n via the Embedding Theorem 2.6 . T o this end, we introduce the following star quiver S : x 1 . . . x m x m +1 . . . x m + p z 0 y 1 . . . y n Let β be the dimension vector of S defined by β ( x ℓ ) = ( 1 if ℓ ∈ [ m ] m if ℓ ∈ { m + 1 , . . . , m + p } , β ( z 0 ) = mp , and β ( y j ) = 1 , ∀ j ∈ [ n ] . W e also need the weight σ a , b of S defined by σ a , b ( x ℓ ) = ( a ℓ if ℓ ∈ [ m ] N if ℓ ∈ { m + 1 , . . . , m + p } , and σ a , b ( z 0 ) = − N , and σ a , b ( y j ) = − b j , ∀ j ∈ [ n ] . Proposition 3.1. Keep the same notation as above. Then (10) T a,b = dim C SI( S , β ) σ a , a . Proof. Following [ CCK25 ], we consider the quiver 6 T : x 1 . . . x m x m +1 . . . x m + p x 0 y 0 y 1 . . . y n Furthermore, for each i ∈ [ m ] , let δ i be the dimension vector defined by δ i ( x 0 ) = p + 1 , δ i ( y 0 ) = p, δ i ( x i ) = 1 , δ i ( x m +1 ) = · · · = δ i ( x m + p ) = 1 , and δ i ( v ) = 0 for all other vertices v ∈ T 0 . It follows from [ CCK25 , Pr oposition 3.4] that E := ( δ 1 , . . . , δ m , e y 1 , . . . , e y n ) is an exceptional sequence of T with T ( E ) = Q p m,n . Now let I : R Q 0 → R T 0 be the transformation ( 8 ) corresponding to T and E . Then the dimension vector b β := I ( 1 ) of T is given by b β ( x i ) = 1 ( i ∈ [ m ]) , b β ( x m + ℓ ) = m ( ℓ ∈ [ p ]) , b β ( x 0 ) = ( p + 1) m, b β ( y 0 ) = pm, b β ( y j ) = 1 ( j ∈ [ n ]) . Next, let us write θ a,b = ⟨ α, −⟩ Q p m,n , where α is the dimension vector of Q p m,n given by α ( x i ) = a i ( i ∈ [ m ]) , α ( y j ) = pN − b j ( j ∈ [ n ]) . Then the dimension vector I ( α ) of T is given by I ( α )( x i ) = a i ( i ∈ [ m ]) , I ( α )( x m + ℓ ) = N ( ℓ ∈ [ p ]) , I ( α )( x 0 ) = ( p + 1) N , I ( α )( y 0 ) = pN , I ( α )( y j ) = pN − b j ( j ∈ [ n ]) . Hence the weight b σ a , b := ⟨ I ( α ) , −⟩ T of T is given by b σ a , b ( x i ) = a i ( i ∈ [ m ]) , b σ a , b ( x m + ℓ ) = N ( ℓ ∈ [ p ]) , b σ a , b ( x 0 ) = 0 , b σ a , b ( y 0 ) = − N , b σ a , b ( y j ) = − b j ( j ∈ [ n ]) . Applying Theorem 2.6 , we obtain that dim SI( Q p m,n , 1 ) θ a , b = dim SI( T , b β ) b σ a , b . 7 Since b σ a,b ( x 0 ) = 0 and b β ( x 0 ) > b β ( y 0 ) , it follows from Theor em 2.1 that dim SI( T , b β ) b σ a , b = dim SI( S , β ) σ a , b . The proof now follows. □ W e are now r eady to prove Theor em 1.1 . For background material relevant to the com- putations in the proof below , we refer the r eader to [ CCK25 , Section 4]. Proof of Theor em 1.1 . W e know from Pr oposition 3.1 that (11) T a , b = dim SI( S , β ) σ a , b . Now , applying ( 7 ) at the source vertices x 1 , . . . , x m + p , we get that (12) dim SI( S , β ) σ a , b = dim SI( e S , e β ) e σ a , b , where e S : x 1 . . . x m x m +1 . . . x m + p z 0 y 1 . . . y n and e β ( x 1 ) = · · · = e β ( x m ) = pm − 1 , e β ( x m +1 ) = · · · = e β ( x m + p ) = m ( p − 1) , e β ( z 0 ) = pm, e β ( y 1 ) = · · · = e β ( y n ) = 1 , and e σ a,b ( x i ) = − a i ( i ∈ [ m ]) , e σ a,b ( x m + ℓ ) = − N ( ℓ ∈ [ p ]) , e σ a,b ( z 0 ) = pN , e σ a,b ( y j ) = − b j ( j ∈ [ n ]) . T o find a closed formula for dim SI( e S , e β ) e σ a , b , we proceed as follows. First, we use Cauchy’s formula to decompose C [rep( e S , e β )] into a direct sum of irreducible repr esen- tations of GL( e β ) . Then, we consider the ring of semi-invariants SI( e S , e β ) := C [rep( e S , e β )] SL( e β ) and sort out those semi-invariants of weight e σ a , b . For convenience, we write V i = C e β ( x i ) , i ∈ [ m + p ] , V = C pm , W j = C e β ( y j ) = C , j ∈ [ n ] . Then C [rep( e S , e β )] = C " m + p Y i =1 Hom C ( V , V i ) × n Y j =1 Hom C ( V , W j ) # = m + p O i =1 S ( V ⊗ V ∗ i ) ⊗ n O j =1 S ( V ⊗ W ∗ j ) 8 = M m + p O i =1 S λ ( i ) V ∗ i ⊗ n O j =1 S µ ( j ) W ∗ j ⊗  S λ (1) V ⊗ · · · ⊗ S λ ( m + p ) V ⊗ S µ (1) V ⊗ · · · ⊗ S µ ( n ) V  , where the direct sum is over all partitions λ ( i ) and µ ( j ) with ℓ ( λ ( i )) ≤ dim V i , ∀ i ∈ [ m + p ] , and ℓ ( µ ( j )) ≤ 1 , ∀ j ∈ [ n ] . Thus, SI( e S , e β ) can be written as M m + p O i =1  S λ ( i ) V ∗ i  SL( V i ) ⊗ n O j =1  S µ ( j ) W ∗ j  SL( W j ) ⊗ m + p O i =1 S λ ( i ) V ⊗ n O j =1 S µ ( j ) V ! SL( V ) . Sorting out those semi-invariants of weight e σ a , b completely determines the partitions λ ( i ) and µ ( j ) . This way we get that SI( e S , e β ) e σ a , b is isomorphic to the weight space of weight e σ a , b ( z 0 ) = pN that occurs in the weight space decomposition of   S ( N p ( m − 1) ) V ⊗ · · · ⊗ S ( N p ( m − 1)) V | {z } p times ⊗ m O i =1 S ( a pm − 1 i ) V ⊗ n O j =1 S b j V   SL( V ) . Thus, SI( e S , e β ) e σ a,b is isomorphic to   S ( N p ( m − 1) ) V ⊗ · · · ⊗ S ( N p ( m − 1) ) V | {z } p times ⊗ m O i =1 S ( a pm − 1 i ) V ⊗ n O j =1 S b j V ⊗ S (( pN ) pm ) V ∗   GL( V ) . The dimension of this weight space is precisely K λ,R . This together with ( 11 ) and ( 12 ) yields the desired formula for T a , b . □ Corollary 3.2. The following formula holds. (13) T a , b = X µ K µ, { ( a 1 ) ,..., ( a m ) , ( N m ) , . . . , ( N m ) | {z } p times } · K µ, { ( b 1 ) ,..., ( b n ) , ( N mp ) } , where the sum is over all partitions µ with at most mp parts. When p = 1 , this recovers the formula for the number of the m × n contingency tables with row and column sums fixed by a and b , as derived from the Robinson-Schenstead-Knuth corr espondence. Proof. W e know from Proposition 3.1 that T a , b = dim SI( S , β ) σ a , b . Using the same strategy as in the proof of Theorem 1.1, we get that SI( S , β ) σ a , b is isomor- phic to   S a 1 V ∗ ⊗ · · · ⊗ S a m V ∗ ⊗ S ( N m ) V ∗ ⊗ · · · ⊗ S ( N m ) V ∗ | {z } p times ⊗ S b 1 V ⊗ · · · ⊗ S b n V ⊗ S ( N mp ) V   GL( V ) , where V = C mp . The dimension of this space is precisely the right hand side of ( 13 ) . □ 9 R E F E R E N C E S [CCK25] C. Chindris, B. Collins, and D. Kline. Hive-type polytopes for quiver multiplicities and the mem- bership problem for quiver moment cones. Algebraic Combinatorics , 8(1):175–199, 2025. [Chi09] C. Chindris. Orbit semigroups and the repr esentation type of quivers. J. Pure Appl. Algebra , 213(7):1418–1429, 2009. [DW00] H. Derksen and J. W eyman. Semi-invariants of quivers and saturation for L ittlewood- R ichardson coefficients. J. Amer . Math. Soc. , 13(3):467–479, 2000. [DW11] H. Derksen and J. W eyman. The combinatorics of quiver representations. Ann. Inst. Fourier (Greno- ble) , 61(3):1061–1131, 2011. U N I V E R S I T Y O F M I S S O U R I - C O L U M B I A , M AT H E M A T I C S D E PA RT M E N T , C O L U M B I A , M O , U S A Email address , Calin Chindris: chindrisc@missouri.edu I N T E R N AT I O N A L C E N T E R F O R M AT H E M AT I C A L S C I E N C E S , I N S T I T U T E O F M AT H E M AT I C S A N D I N F O R - M AT I C S , B U L G A R I A N A C A D E M Y O F S C I E N C E S , A C A D . G . B O N C H E V S T R . , B L . 8 , S O FI A 1 1 1 3 , B U L G A R I A Email address , Deepanshu Prasad: deepanshu.prasad@gmail.com 10