Pattern avoidance in labelled trees

P A T TERN A VOIDANCE IN LABELLED TREES VLADIMIR DOTSENKO A bstract . W e discuss a new notion of p a ttern avoida nce moti- vated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation o f various types of consecutive pattern avoidance studied before: consecutive patterns in words, permu- tations, colour ed permutations etc. The notion of W ilf equivalence for patterns in per mutations admits a straightforward generalisa- tion for (sets of) tree patterns; we describe classes for trees with small numbers of leaves, and give sev e ral bijections between trees avoiding pattern sets f rom the same cla ss. W e also e xplain a few general results for tree pattern avoidance, both f or the exact and the asymptotic enumeration. 1. I ntroduction For pattern avoidance in words, a p a rt fr om the “real wor d interpr e- tation” (enumerate wor ds not containing any obscene subwords), the pattern avoidance pr oblem arises fr om studying (noncommutative) algebras with monomial r elatio ns. For example, describing words in the alphabet { A , B , . . . , Z } not containing the word FCUK as a sub- wor d is equivalent to figuring out which monomials in generators x 1 , . . . , x 26 form a basis in the algebra k h x 1 , . . . , x 26 | x 6 x 3 x 21 x 11 = 0 i . The significance of algebras with monomial relations is, in turn, ex- plained by the theory of Gr ¨ obner bases which gives a method of finding a “monomial replacement” for every algebra with monomial r elations [40]. Similarly , consecutive pattern a voidance in permu- tations [2, 3] and colour ed p ermutations [29] can be interpreted in terms of shu ffl e a lgebras with monomial relatio ns [9]. The goal of this paper is to intr oduce to the combinatorics audience a new no- tion of pattern avoidance naturally arising when studying operads. Operads are similar to associative a lgebras, but while associative al- gebras a nd gr oups capture the kind of associativity that one observes when composing transformations of some set, operads captur e the associativity exhibited when composing operations with several ar- guments. The pr operty of an operation of having more than one ar gument results in a choice that is not present in the choice of alge- bras: one ma y either assume that our operations do not possess any symmetries or allow symmetries in the picture. In the former case, the pattern avoidance in question is the pattern avoidance in planar r ooted tr ees; a few pa pers, both of combinatorial spirit [1, 33, 3 6, 1 3 ] 1 2 VLADIMIR DOTSENKO and operads-inspir ed [28] have been dealing with the arising ques- tions of enumeration. I n the case of operations with symmetries, the corr esponding notion ha s not been studied befor e, and this paper is an attempt to give an ele mentary intr oduction to the arising re- sear ch ar ea. The “right” notion of a monomial relation for operations with symmetries is not as obvious as one might think: the action of symmetries makes every relation have too many consequences, and the arising class of “operads with monomial r elations” appears to be way too narrow to be truly interesting or useful. The way to define monomial relations which avoids narro wing things down, which in particular led to a theory of operadic Gr ¨ obner bases, was suggested in [8]; the corr esponding algebraic object is called a shu ffl e operad. I n this paper we, however , shall try to concentrate on the combinatorial aspects of the subject, touching the algebraic aspects only briefly; for details on the algebraic a spects, see, for example, [6, 7, 8]. Though we attempt to keep this article relatively self-contained, familiarity with the key result s of the theory of consecutive pattern avoidance in permutations could be useful; for relevant historical information on pattern avoidance as well as the state-of-art of this area the reader is referr ed to a recent monograph [24]. Ther e are several types of questions in the theory of tree patterns which ar e mea ningful from the operadic viewpoint. First of all, for a given set of patterns, e xact enumeration results for tr ees avoiding that set are very important. Some examples of that sort appear in the following sections; in many cases the corr esponding se q uences of numbers are well known in combinatorics, but in many other cases one ends up with a sequences that appear to be unrelated to classical enumeration problems. Second, ther e is a question of asymptotic enumeration. W e shall pro ve some results of that sort r elying on the Golod–Shafar evich technique [17] (which has recently been re- discover ed in r elation to combinatorics of pattern avoidance [4, 5, 35]). Also, ther e is a question of recog nising the class of generating functions arising in this type of enumerating questions. However , while for wor d avoidance the answer is that the generating functions arising as answers for enumerating the avoidance of finite sets of wor ds are always rational, even for consecutive pattern avoidance in permutations the class of arising generating functions does not have a satisfactory description. The a nswer is known if one ignores the leaf la be ls completely; in that case, the generating functions are algebraic (as proved in [36] in the case of binary tr ees, in [13] in the case of ternary tr ees, a nd in [22 ] in the general case). For the notion of tree pattern avoidance discussed here, it follows from one of the theorems of [22] that under some additional assumptions on the set of pa tterns (“shu ffl e regularit y”) the generating functions for P A TTERN A VOIDANCE IN LABEL LED TREES 3 tr ee pattern avoidance a r e di ff er entially algebraic (i.e. satisfy non- linear di ff erential eq ua tions with polynomial coe ffi cients). Finally , it is interesting to e numerate the W ilf equivalence classes of sets of tr ee patterns 1 , a nd we discuss some basic results of that sort. This paper is organised a s follows. In Section 2, we define planar labelled ro oted tr ees and tr ee pa tterns, an d show that this notion includes the classical types of consecutive pattern avoidance as par- ticular cases. I n Section 3, we present some results on asymptot ics for tr ee pattern avoidance, and introduce the notion of growt h rate for a given set of patterns. In Section 4, we formulate an exact enume ration r esult on tree pa ttern avoidance which follows from our work with Khor oshkin [7], and discuss some consequences of that result and r elated questions. Finally , in Section 5 we discuss various examples of patterns with small numbers of leaves. Each of these sections also contains some conjectur es and natural questions that are beyond the scope of this paper . W e also included an appendix e xplaining how pattern avoidance in tr ees arises in the operadic context. Acknowledgements. I wish to thank Anton Khoro shkin and Dmitri Piontkov ski for sending me a copy of their forthcoming pr eprint [22 ]. 2. P lanar labelled tree p a tterns 2.1. T ree patterns. T rees have ver ti c es and edge s . A rooted tree is a tr ee with a distinguished vertex, called the root . A rooted tree can be directed “awa y from the root ”; this way every vertex except for the r oot has exactly one p ar ent . V ertices whose common parent is a given vertex v are called c hildren of v . V ertices that have at least one child are called internal , vertices with no children ar e called leaves . A planar rooted tree is a ro oted tree together with a total order on the set of children of each vertex, we shall think of it as of embedded in plane, the childr en of each vertex placed in the increasing or der fr om the left to the right. Thr oughout the paper , X = F n ≥ 2 X n is a finite a lphabet r epr esented as a disjoint union of its subsets X n , n ≥ 2 . Definition 1. A planar X-labell ed rooted tree is a planar rooted tree T with no internal vertices having exactly one child, and with a la - belling of all vertices fulfilling the following rest rictions: - every internal vertex v with m children is labelled by an ele- ment x v ∈ X m ; - every leaf of T is labelled by a positive integer in such a wa y that the following two conditions are satisfied: 1 Equivalence classes f or “ strong equivalence” might lead to interesting combi- natorial results a s well, but are much less natural from the oper a d point of view . 4 VLADIMIR DOTSENKO (1) (labelling set condition) the leaf labe ls are in one-to-one corr espondence with the set [ l ] = { 1 , 2 , . . . , l } (where l is the number of leaves of T ); (2) (local increasing condition) if we temporarily assign to each internal vertex v the smallest of the labels of leaves that ar e descendants of v in T (thus every vertex of T , should it be an internal vertex or a leaf, has an integer assigned to it), then for each internal vertex i the integers assigned to its childr en increase from the left to the right. Notation: for i ≥ 0, l ≥ 1 we denote by LT i , l ( X ) the set of planar X -labelled r ooted tr ees with i internal vertices and l le aves, (1) LT l ( X ) = [ i ≥ 0 LT i , l ( X ) is the set of planar X -labelled r ooted trees with l leaves, (2) LT ( X ) = [ i ≥ 0 , l ≥ 1 LT i , l ( X ) is the set of all planar X -labelled ro oted tr ees. In particular , L T 0 , l ( X ) is non-empty only for l = 1 (and in that case it consists of exactly one element, a one-vertex tree with no edges), and LT 1 ( X ) = LT 0 , 1 ( X ) (since every internal vertex has at least two children, hence in the pr esence of an internal vertex we e n d up with at le ast two leaves). Example 1 . Let X = X 2 = {◦ , •} . The following tree is in LT 7 , 8 ( X ): (3) • ◦ • ◦ ◦ ◦ ◦ 1 3 2 7 4 6 5 8 . A tree T is said to be a left (right) c om b if for e very internal vertex of T only its leftmost (rightmost) child may not be a leaf. N ote that the local increasing condition makes the notions of “left” and “right” very di ff e r ent: for example, if a planar ro oted tr ee t is a left comb, the only rest riction on the leaf labels is that the leftmost leaf of t is labelled by 1; by contrast, if t is a right comb, there is only one leaf labelling satisfying the local incr easing condition. Most of the time throug hout the paper we consider only the case X = X d , thus assuming that for our trees all internal vertices have the same number of childr en (that is, d childr en). This a ssumption is mostly technical (it allows for closed formulas in various statements), in particular , all the asymptotic results we prove and conjectur e in Section 3 are expected to be true in full generality . One particular simplification is that for such tr ees the number of internal vertices and P A TTERN A VOIDANCE IN LABEL LED TREES 5 the number of leaves are related: every tree with k internal vertices has kd − k + 1 leaves. Let us prove a basic e nume rative result which will be useful later . Proposition 1. For X = X d , w e have (4) |LT kd − k + 1 ( X ) | = | X d | k ( kd )! ( d !) k · k ! . Proo f. It would be useful for our purposes to consider , along with planar X -labelle d root ed tr ees, two other types of tr ees. W e denote by T n ( X ) the set of planar r ooted trees with n leaves whose internal vertices ar e labelled by X , and by T n ( X ) the set of planar ro oted tr ees with n leaves whose in ternal vertices are labelled by X , and whose leaves ar e la be lled by { 1 , 2 , . . . , n } (in all possible ways). Recall that for X = X d = {•} , we have [39] (5) | T kd − k + 1 ( X ) | = 1 kd − k + 1 kd k ! . For the general case, we note that | T kd − k + 1 ( X ) | is | X d | k times larger , since every of k internal vertices should acquire a label from X d . Also, it is clear that (6) |LT kd − k + 1 ( X ) | = 1 ( d !) k |T kd − k + 1 ( X ) | , since for each of the k internal vertices of a tr ee only one of the d ! pe rmutations of its subtrees fulfils the local incr easing condition. Finally , (7) |T kd − k + 1 ( X ) | = ( kd − k + 1)! | T kd − k + 1 ( X ) | , since all leaf labelling ar e allowed in T kd − k + 1 ( X ), a nd the formula follows.  By a subtr ee of a pla nar labelled r ooted tr ee T we always mean a subtr ee S with its root at one of the internal vertices of T such that for each internal vertex of S all its children in T are a lso its children in S . (This way we can guarantee that the labels of internal vertices make sense for S .) Note that the second condition on leaf-labelling assigns to each internal vertex of a tree T a temporary integer label, so that a subtr ee S of a tree T almost be longs to LT ( X ): its le aves are labe lled by integers such that the local increasing condition is satisfied (but the labelling set condition might not be satisfied). Re placing, for a subtr ee S with l lea ves, its leaf labels by 1 , 2 , . . . , l in the uniq ue order - pr eserving way , we shall obtain a tree st( S ) ∈ LT ( X ) which we call the standardisation of S . 6 VLADIMIR DOTSENKO Definition 2. A tree T ∈ L T ( X ) is said to contain a tree P ∈ L T ( X ) as a pattern if ther e exists a subtr ee S of T for which P = st( S ). Otherwise T is said to avoid S . Example 2. Let us recall the tr ee fr om Example 1, and consider its subtr ee r epr esented by thick lines in the following figure: (8) • ◦ • ◦ ◦ ◦ ◦ 1 3 2 7 4 6 5 8 . This subtree a cquires the le af numbering ◦ • ◦ 1 5 2 4 , and a fter standard- isation we get ◦ • ◦ 1 4 2 3 . So ◦ • ◦ 1 4 2 3 occurs in our tr ee as a pattern. Thr oughout the paper , we only consider one type of tr ee patterns, so we often use the wor ds “tree pattern” where one should say “ p la- nar labelled r ooted tr ee pattern”; we hope that the reader will appre- ciate this attempt to abbr eviate things. Let us fix some set of labels X , and consider the pattern avoidance for planar X -labelled trees. The central question arising in the theory of pattern avoidance is that of enumeration of objects that a void the given set of forbidden p a tterns P or , mor e generally , that contain exactly d occurr ences of patterns from P . This question naturally leads to the following equivalence relations for tree patterns. T wo sets of tr ee patterns P , P ′ ⊂ LT ( X ) are said to be W ilf equivalent (notation: P ∼ W P ′ ) if for every l , the number of P -avoiding tr ees with l leaves is equal to the number of P ′ -avoiding tr ees with l leaves. (In the case of (non-consecutiv e) pe rmutation patterns the same notion was introduced by W ilf [42].) Mor e gene rally , P and P ′ ar e said to be (str ongly) equivalent (notation: P ∼ P ′ ) if for every l and every k ≥ 0, the number of tr ees with l leaves that ha ve k occurr ences of patterns from P is e qual to the number of tr e e s with l leaves that have k occurr ences of patterns from P ′ . For enumeration, we shall primarily use the exponential generat- ing functions with respect to the number of leaves in tr e es, so that, for example , the generating functions for the label set X a nd the pattern set P a re f X ( z ) = P n ≥ 1 | X n | z n n ! and f P ( z ) = P l ≥ 1 | P ∩LT l ( X ) | z l l ! r espec- tively . W e shall denote the set of all tr ees avoiding the pattern set P by L T no- P ( X ), a nd its subset consisting of trees with l leaves by LT l , no- P ( X ). The corr esponding generating function is denoted by P A TTERN A VOIDANCE IN LABEL LED TREES 7 f no- P ( z ): (9) f no- P ( z ) = X l ≥ 1 |LT l , no- P ( X ) | z l l ! . Remark 1. One can also include the second variable in the generating series to count the internal vertices separately , and use the series (10) g no- P ( z , t ) = X i , l ≥ 1 |LT i , l , no- P ( X ) | t i z l l ! , which in particular would make all the sets finite when internal ver - tices of our trees are allowed to have a single child (and the label set X includes X 1 ). T o keep the exposition simple, we avoid discussing these subtleties here. The key feature of exponential generating functions in the con- text of pla nar X -labelled root ed trees is expr essed by the following pr oposition. Proposition 2. Suppose that K and L are two sets of planar X-labelled roo ted trees. Let us d e fine a set M as foll ows: it consis ts of all trees T that have an occ urren ce of a tree pattern from K rooted at the root of T , with the additional condition that all the subtr ees rooted at the leaves of that pattern are occurrences of tree patterns f rom L . Then (11) f M ( z ) = f K ( f L ( z )) . Proo f. Clearly , (12) f K ( f L ( z )) = X l ≥ 1 | K ∩ L T l ( X ) | ( f L ( z )) l l ! , and it r emains to note that the coe ffi cient of z n in ( f L ( z )) l is the number of order ed for e sts of l tr ee patterns fro m L with the total leaf set { 1 , . . . , n } , ther efor e ( f L ( z )) l l ! can be thoug ht of as the enume rator for forests satisfying the increasing condition for minimal lea ves.  2.2. T ree patterns and other types of consecutive patterns. In this section we assume, for simplicity , that X = X 2 (this correspo nds to considering only binary tr ee patterns). Our first observation is that for | X | = 1 the set of all permuta- tions is naturally embedded in L T ( X ) as left combs: recall that a left comb has no conditions on where to put labels 2 , 3 . . . , so left combs with n + 1 leaves are in one-to one correspondence with permuta- tions of length n . If we denote by T ( σ ) the tree corr esponding to the permutation σ , then subtrees of T ( σ ) are in one-to-one corr espon- dence with subwords of σ , and the notion of a tree pattern for left combs corr esponds pr ecisely to the notion of a consecutive pattern for permutations. Moreover , if Π is a set of consecutive permutation 8 VLADIMIR DOTSENKO patterns, and P Π contains the left combs correspo nding to elements of Π and the right comb with thr e e leaves, then the number of tr ee s with n + 1 le aves that avoid P Π is equal to the number of p ermu- tations of length n that avoid Π . For | X | > 1, the same construc tion with left combs leads naturally to the notion of pattern avoidance in colour ed p e rmutations [29]. Mor eover , the set of all words in a given alphabet A is naturally embedded in LT ( X ) with X = X 2 = A as right combs. Indeed, re- call that for a right comb with n + 1 leaves there is exactly one way to label its leaves to fulfil the local increasing condition; to obtain a planar X -labelled ro oted tree, it remains to label its internal vertices by A , and the ways to do so are in one-to-one corr espondence with A -wor ds of length n . If we denote by T ( w ) the tree corr esponding to the wor d w , then subtrees of T ( w ) are in one-to-one corr e spondence with subwor ds of w , and the notion of a tree pattern for right combs corr esponds pr e cisely to the notion of a divisor for words. Moreover , if W is a set of words, and P W consists of the right combs corre- sponding to eleme n ts of W and all the left combs with thr ee le aves, then the number of tr ees with n + 1 leaves that avoid P W is eq ual to the number of words of length n without divisors from W . It is also possible to go the other way round and r eplace trees by objects resembling p ermutations and wor ds. Let us assume, as above, that | X | = 1 (ignoring this technical assumption will, as al- ways, me rely lead to coloured objects of the same sort). Ther e is a very natural way to “straighten” the tree patterns and thus translate our questions into similar questions about patterns in sequences. R e- call that the total number of planar labe lle d r ooted tree patterns in our case is equal to (2 n )! 2 n n ! = (2 n − 1)!! , the double factorial number . This number is also equal [14] to the number of permutations of the mul- tiset { 1 , 1 , 2 , 2 , . . . , n , n } for which all the numbers app e aring between the two occurr ences of k are gr eater than k (for e very k = 1 , . . . , n ). T o a planar labelled r ooted bina ry tree T , it is ea sy to assign recursively a permutation σ ( T ) of that sort. For that, it is convenient to think of T as of a left comb with subtr ees grafted in the places of right children of internal vertices. W e denote those subtr ees by T 1 , . . . , T k , in the or de r fro m the leftmost one to the rightmost one. Each subtree T i has its leaf labels belonging to a subset of { 1 , 2 , . . . , n } , so, strictly speak- ing, they a re not trees of the sort we consider , but, as usual, we can identify them with planar labelled ro oted trees via standardisation, so we may apply σ to them, obtaining permutations of appropriate multisets. W e assume that σ takes the only one-vertex tr e e to the empty word, a nd put (13) σ ( T ) = ( 1 σ ( T 1 )1 for k = 1 , st( σ ( T 1 ) σ ( T 2 ) · · · σ ( T k )) for k ≥ 2 . P A TTERN A VOIDANCE IN LABEL LED TREES 9 For example, σ ( 1 2 ) = 11, σ ( 1 2 3 4 ) = 112332, σ ( 4 1 2 3 ) = 122133, σ ( 1 2 3 4 ) = 1 22331; this leads to a me aningful notion of a generalised permutation pattern. It would be interesting to investigate this no- tion pro perly , in particular , to explor e the links with patterns in set partitions [19 , 25, 37], and a lso to see if the constructio ns of [10] can be adapted here. 3. A symptotic en umera tion In this section, we discuss results on the asymptotic enumeration of trees avoiding a given set of patterns, where the result s turn out to be in a wa y mimicking the result s on the asymptotic enumeration for consecutive patterns in pe rmutations [9, 11 ]. Our main tool is the following r esult, an adap tation of the classical Golod–Shafarevich in- equality [17]; it is closely r ela ted to a similar ineq ua lity for symmetric operads [34]. Theorem 1. For every (possibly infinite) pattern s e t P , we have the fol - lowing coe ffi cient-wise inequality of power series: (14) f P ( f no - P ( z )) − f X ( f no - P ( z )) + f no - P ( z ) ≥ z . Proo f. Let us consider two series of finite sets: the set B n is the subset of LT n ( X ) consisting of trees T whose subtr e es root e d at the children of the root of T avoid patterns from P , a nd the set C n is the subset of the set of pairs P × L T n ( X ) consisting of all p a irs ( P , T ) where there exists a subtr ee S of T ro oted at the roo t of T for which st( S ) = P , and all the trees ro oted at the leaves of S avoid patterns from P . Pr oposition 2 implies that X n ≥ 1 | B n | n ! z n = f X ( f no- P ( z )) , (15) X n ≥ 1 | C n | n ! z n = f P ( f no- P ( z )) . (16) Ther efor e, our power series inequality translates into (17) | C n | − | B n | + |LT n , no- P ( X ) | ≥ 0 , n ≥ 2 . This follows fr om an observation that there e xists an obvious surjec- tion from C n ⊔ LT n , no- P ( X ) onto B n : a tr ee fro m B n either avoids P , or has a pattern fr om P r ooted at its roo t.  10 VLADIMIR DOTSENKO Corollary 1. Suppose that the power se r ies h ( z ) = 1 1 − f X ( z ) z + f P ( z ) z has non- negative coe ffi cients. Then we have (18) |LT n , no - P ( X ) | n ! ≥ 1 n [ z n − 1 ] h ( z ) n . Proo f. Let g ( z ) = z − f X ( z ) + f P ( z ). A ccor ding to the Lagrange inversion formula [39], for the coe ffi cients of the compositional inverse g h− 1 i ( z ) we have (19) [ z n ] g h− 1 i ( z ) = 1 n [ z n − 1 ] z g ( z ) ! n = 1 n [ z n − 1 ] h ( z ) n , so under our assumption on h ( z ) the power series g h− 1 i ( z ) has non- negative coe ffi cients. According to Theor em 1, the power series (20) f P ( f no- P ( z )) − f X ( f no- P ( z )) + f no- P ( z ) = g ( f no- P ( z )) has non-negative coe ffi cients as well, so we see that (21) f no- P ( z ) = g h− 1 i ( g ( f no- P ( z ))) = = X n ≥ 1  1 n [ z n − 1 ] h ( z ) n  g ( f no- P ( z )) n ≥ X n ≥ 1  1 n [ z n − 1 ] h ( z ) n  z n . This mea ns that for every n ≥ 1 we have (22) |LT n , no- P ( X ) | n ! ≥ 1 n [ z n − 1 ] h ( z ) n , which is exactly what we want to prov e.  W e use this result to pr ove the following theorem which gives a se- ries of examples when the set of P -avoiding trees with n leaves grow s rapidly , nam e ly as C n times the number of all tr ees with n lea ves for some constant C . A part of the pr oof is parallel to the corr esponding pr oof in [9]. Theorem 2. Suppose that X = X d for s om e d ≥ 2 , and that the power series (23) h ( z ) = 1 − | X d | z d − 1 d ! + f P ( z ) z has a root α > 0 . T h en we have (24) |LT n , no - P ( X ) | ≥ ( d !) 1 d − 1 | X d | 1 d − 1 α ! n − 1 |LT n ( X ) | . Proo f. Recall that if X = X d , then the set L T n ( X ) i s non-empty only for n = kd − k + 1 = k ( d − 1) + 1 for some k . Therefor e, h ( z ) is a power series in z d − 1 ; we shall consider the series (25) ˆ h ( t ) = 1 − t + X t ≥ 2 a k t k , P A TTERN A VOIDANCE IN LABEL LED TREES 11 for which h ( z ) = ˆ h  | X d | d ! z d − 1  ; the only fa ct about the coe ffi cients of that series that we use is that all the coe ffi cien ts a k ar e non-negative ( a k is a positive multiple of the number of certain labelled tr e e s with k internal vertices). Under our assumption, this power series has a r oot β = | X d | d ! α d − 1 . Let us consider multiplicative inverse, P l ≥ 0 b l t l : = ( ˆ h ( t )) − 1 ; clearly , b 0 = 1 and b n − b n − 1 + P n k = 2 a k b n − k = 0. Let us prove by induction that b n ≥ β − 1 b n − 1 . Indee d , for n = 1 this statement is obvious ( β ≥ 1 because otherwise ˆ h ( β ) is evidently positive), and for n > 1 we note that by the induction hypothesis b n − 1 ≥ β 1 − k b n − k , so b n = b n − 1 − n X k = 2 a k b n − k ≥ b n − 1 − n X k = 2 a k β k − 1 b n − 1 ≥ ≥ b n − 1 − X k ≥ 2 a k β k − 1 b n − 1 = β − 1 b n − 1        β − X k ≥ 2 a k β k        = β − 1 b n − 1 , and the statement follows. Hence the k th coe ffi cient of ( ˆ h ( t )) − 1 is gr ea ter than or equal to β − k . Ther efore, the ( k ( d − 1)) th coe ffi cient of h ( z ) is gr e a ter than or equal to  | X d | d !  ·  | X d | α d − 1 d !  − k = α − k ( d − 1) . This means that (26) h ( z ) ≥ X k ≥ 1 α − k ( d − 1) z k ( d − 1) = 1 −  z α  d − 1 ! − 1 , so consequently (27) h ( z ) n ≥ 1 −  z α  d − 1 ! − n . Since the coe ffi cients of h ( z ) are non-negative, Corollary 1 applie s, and we deduce that (28) |LT kd − k + 1 , no - P ( X ) | ( kd − k + 1)! ≥ 1 kd − k + 1 [ z k ( d − 1) ] h ( z ) kd − k + 1 ≥ ≥ 1 kd − k + 1 α − k ( d − 1) kd − k + 1 + k − 1 k ! = 1 kd − k + 1 α − k ( d − 1) kd k ! . This, in the view of Formula (4), simplifies to (29) |LT kd − k + 1 , no - P ( X ) | ≥ ( kd − k + 1)! 1 kd − k + 1 α − k ( d − 1) kd k ! = = α − k ( d − 1) ( kd )! k ! = d ! α d − 1 ! k ( kd )! k ! d ! k = d ! | X d | α d − 1 ! k |LT kd − k + 1 ( X ) | , which easily can be transformed into the inequality we want to pr ove.  12 VLADIMIR DOTSENKO This theorem, in particular , can be used to obtain estimates in the case of one tree pattern in the case | X | = 1, that is for tr ees with all internal vertices of the same arity and carrying the same labe l. Theorem 3. Suppose that | X | = 1 , so that X coincides with a one-elem e nt set X d for some d ≥ 2 , and that the set of forbidden patterns consis ts of one single p attern P with k ≥ 2 internal v ertices. Then for every pair ( d , k ) except for (2 , 2) , (2 , 3) , (2 , 4) , and (3 , 2) th e re exists a positive number C (depending only on d and k but not on the actual pattern) such that (30) |LT n , no -P ( X ) | ≥ C n − 1 |LT n ( X ) | . Proo f. It is enough to show that for all p airs ( d , k ) except for (2 , 2), (2 , 3 ), (2 , 4), and (3 , 2), the polynomial h ( z ) = 1 − z d − 1 d ! + z k ( d − 1) ( kd − k + 1)! has a positive root . Denoting, as above, t = z d − 1 d ! , we see that it is enough to pr ove that the polynomial ˆ h ( t ) = 1 − t + d ! k t k ( kd − k + 1)! has a positive r oot. For that, we note that the derivative − 1 + kd ! k t k − 1 ( kd − k + 1)! of the p olynomial ˆ h ( t ) has the only positive ro ot t 0 =  ( kd − k + 1)! kd ! k  1 k − 1 ; thus, to prov e that ˆ h ( t ) has a p ositive root , it is enough to pro ve that its minimal value is attained at t 0 and is negative. Using the formula t k − 1 0 = ( kd − k + 1)! kd ! k , we see that ˆ h ( t 0 ) = 1 − t 0 + t 0 k , so it su ffi ces to pro ve that t 0 > k k − 1 , or (31) ( kd − k + 1)!( k − 1) k − 1 d ! k k k > 1 , which is true in all cases we consider , since it is true for ( d , k ) = (2 , 5), ( d , k ) = (3 , 3), and ( d , k ) = (4 , 2), and its left hand side increases if either d or k increases.  Our r esults suggest a new numerical invariant of a set of pa tterns: Definition 3. The gro wth r ate of a set of pa tterns P ⊂ LT ( X ) is (32) lim sup n →∞ |LT n , no- P ( X ) | |LT n ( X ) | ! 1 n − 1 . W e expect that the methods of this section can be easily gene ralised to prov e that for e very set of la be ls X (and for some k depending on X ) every pattern in LT ( X ) with at least k in ternal vertices has positive gr owth. Mor e over , computer experiments suggest that the follow- ing conjectur e generalising W arlimont’s conjecture for consectuive patterns [41] (proved r ecently by Ehr enbor g, Kitaev and Perry [10] via a beautiful link between consecutive pattern avoidance and the spectral theory for integral operators on the unit cube) holds for tree patterns. P A TTERN A VOIDANCE IN LABEL LED TREES 13 Conjecture 1. For every set of l abels X , there exists an integer d that for every pattern P in LT ( X ) wi th at le ast d internal vertic e s and som e numbers c ( P ) > 0 , λ ( P ) > 1 we have (33) |LT n , no -P ( X ) | |LT n ( X ) | ∼ c ( P ) λ ( P ) − n . It would be most interesting to adapt the approach of [10 ] for tr ee patterns; such an adaptation, in ad d ition to its consequences for the asymptotic enumeration questions, may be of substantial interest for the operad theory as well. Another natural question arising from the asymptotic enumeration is to understand the “gr owth hierarchies” of tree patterns, using the strategy of [26] or otherwise. 4. E xact enumera tion : cluster inversion formula The following result on power series inversion is simultaneously a generalisation of the inversion formula for planar tr ee p a tterns [1, 28, 33] and of the cluster inversion formula of Goulden and Jackson for words and pe rmutations [20, 32]. This result is pro ved in [7] by simple homological algebra; we formulate it here in a di ff erent way , so that an interested r eader will easily pr ove it direct ly using the inclusion-exclusio n formula, similarly to the usual cluster inversion. Definition 4. Let P be a set of patterns in LT ( X ). A tree T together with a collection of its subtrees T 1 , . . . , T k is said to be a k-cluster if the following two conditions hold: (1) For every i , the subtr ee T i is an occurr ence of a pattern from P : st( T i ) ∈ P , (2) for every edge e of T that joins two internal vertices, it is an edge between the two internal vertices of some T i . By de finition, the set of 0-clusters is the set of labe ls X . Informally , a k -cluster is a tree which is completely cover ed by k copies of patterns from P . Let us denote by c n , k ( P ) the number of k -clusters for which the tr ee T has n leaves. Theorem 4 ([7]) . The comp ositional inver s e f or f no - P ( z ) can be c omputed via clusters as follows: (34) f h− 1 i no - P ( z ) = z − X n ≥ 1 , k ≥ 0 ( − 1) k c n , k ( P ) z n n ! . The following consequence of the gene ral inversion formula is a direct analogue for planar labelle d tree patterns of the inversion formula [1, 28, 33] for plana r tree patterns. 14 VLADIMIR DOTSENKO Corollary 2 . Suppose that X = X 2 , and that P and Q are two com- plementary sets of tree patterns with 3 leaves, P ⊔ Q = LT 3 ( X ) . The corresponding generating functions for pattern avoidance are i nver se to each other: (35) f no - P ( − f no - Q ( − z )) = f no - Q ( − f no - P ( − z )) = z . Proo f. It is ea sy to see that in this case k -clusters for P are in one-to- one corr espondence with tr ees that avoid Q (each such tr ee admits the unique covering by patterns from P ), and that the signs in the inversion formula match those suggested by (35 ) (since in our case the unde rlying tree of every n -cluster has n + 2 leaves).  Note that for left (right) combs corresponding to some consecutive permutation patterns (words), clusters for left combs are the left (right) combs correspo nding to the usual Goulden–Jackson clusters for these patterns (wor ds). This instantly pr oves the following result. Corollary 3. Suppose that two se ts of consecutive permutation patterns (words) are Wilf equivalent. Then the two sets of tree patterns consisting of the left (right) c ombs corresponding to the gi ven perm utation patterns (words) ar e Wilf e quivale nt as tr ee patterns. As in the case of pe rmutations, we expect that at least in the case of a single pattern a careful study of its self-overlaps (“overlap sets” of [23], or equivalently “ overlap maps” of [31]) would be very beneficial for studying W ilf equivalence. W e shall discuss it in de tail elsewhere, mentioning a particular case briefly in the next section. Let us conclude this section with an open question. For consecu- tive patterns in permutations, Mendes a nd Remme l developed the symmetric functions method [30] for enumeration of pe rmutations avoiding the given set of patterns. It is natural to expect that this method can be generalised to de al with the case of tr ee pa tterns, pos- sibly making use of the plethysm for symmetric functions wher e our formulas compute compositions of power series. Some inversion for- mulas in the completion of the algebra of symmetric functions e xist in the operadic context, being p rovided by homological algebra, in particular the operadic Koszul duality for symmetric operads [16], however , once we move fro m abstract trees to their r epr esentatives (that is, pla nar labelled ro oted tr ees studied in this paper), ther e is no clear way to incorporate symmetric functions in the pictur e. 5. E xamples 5.1. Case X = X 2 , | X | = 1 . In this section, we assume that X = X 2 , | X | = 1, that is, we only work with binary trees, a nd do not use labels for internal vertices. T o simplify the notation, we suppress X in the formulas, a nd write simply LT n etc. P A TTERN A VOIDANCE IN LABEL LED TREES 15 W e begin with classifying the W ilf classes of pattern sets with three leaves. Theorem 5. There exi st exactly f our Wilf classes of sets of pattern sets with three leaves. Proo f. This theorem follows fr om the two lemma s which give a pre- cise de scription of the five W ilf classes. Lemma 5.1. The three patterns 2 1 3 , 3 1 2 , 3 2 1 are W ilf equivalent to each other; the number of trees with n leaves avoiding either of them is equal to ( n − 1)! for each n ≥ 3 . Proo f. Let us denote P 1 = 2 1 3 , P 2 = 3 1 2 , a nd P 3 = 3 2 1 . A correspondence ρ between the set of P 1 -avoiding tr ees and the set of P 2 -avoiding trees can be defined recur sively as follows. By definition, ρ ( 1 2 ) = 1 2 . Let us r e pr e sent a tree T ∈ LT n , no- P 1 as a left comb with some subtr ees T 1 , . . . , T k (listed along the way from the root) grafted at its “right-look ing” leaves. T o determine ρ ( T ), we apply ρ to the subtr e e s T i and reverse the or der of grafting. In other wor ds, we graft ρ ( T k ) in the place of T 1 , ρ ( T k − 1 ) in the place of T 2 etc. Clearly , ρ that identifies L T n , no- P 1 with LT n , no- P 2 . A correspo ndence κ between the set of P 1 -avoiding tr ees and the set of P 3 -avoiding tr ees can be defined r ecursively as well. By definition, ρ ( 1 2 ) = 1 2 . Let us repr e se nt a tr ee T ∈ LT n , no- P 1 as a left comb with some subtr ees T 1 , . . . , T k (listed along the way fro m the ro ot) grafted at its right-look in g leaves. W e note that the set LT n , no- P 3 is the set of all left combs with n leaves; by induction we may assume that we already know the left combs κ ( T 1 ), . . . , κ ( T k ). L e t κ ( T ) be the left comb whose right-looking leaves, listed along the way from the r oot, are the right-loo king leaves of κ ( T k ), the right-look ing leaves of κ ( T k − 1 ), . . . , the right -looking leaves of κ ( T 1 ). The observation (which can easily be pro ved by induction) that the label of the right-looking leaf of κ ( T ) which is the farthest from the root is the equal to the smallest leaf label of T 1 shows how to construct the inverse of κ , so we identified LT n , no- P 1 with LT n , no- P 3 . In a d d ition, since the set L T n , no- P 3 is the set of all left combs with n leaves, it has the cardinality ( n − 1)!, so for each of the three subsets P ⊂ LT 3 with | P | = 1 and for each n , there are exactly ( n − 1)! di ff erent P -avoiding tree with n leaves.  16 VLADIMIR DOTSENKO Lemma 5.2. The three two-pattern sets { 3 1 2 , 2 1 3 } , { 3 1 2 , 3 2 1 } , and { 2 1 3 , 3 2 1 } are Wilf equivalent to each other; th e number of trees with n leaves avoid i ng either of these sets is equal to 1 for each n ≥ 3 . Proo f. Indeed , for P = { 3 1 2 , 2 1 3 } the only P -avoiding tree with n leaves is the only right comb, for P = { 3 1 2 , 3 2 1 } the only P - avoiding tree with n leaves is the only left comb with labels of right- looking le aves increasing along the way from the root , and for P = { 2 1 3 , 3 2 1 } the only P -avoiding tree with n leaves is the only le ft comb with labels of right-looking leaves decr easing a long the way fr om the root. Therefor e for each of the three subsets P ⊂ LT 3 with | P | = 2 and for each n , there is exactly one P -avoiding tr e e with n leaves. Alternatively , one can apply the inversion formula (35 ): fro m Lemma 5.1, we conclude that the exponential gene rating function for every one-pattern set is equal to − log(1 − z ); computing its in- verse and a djusting the signs instantly shows that the exponential generating function for every two-pattern set is e xp( z ) − 1, which is the exponential generating function for the sequence 1 , 1 , 1 , . . . .  Since the empty p attern set and the pattern set containing all tr e es with three leaves form their own W ilf classes, the theor e m follows.  For pattern sets with at le ast four leaves, we have only pa rtial r esults. Note that ther e are 15 patterns with four leaves: 4 3 1 2 , 3 4 1 2 , 4 2 1 3 , 2 4 1 3 , 3 2 1 4 , 2 3 1 4 , 4 1 2 3 , 3 1 2 4 , 2 1 3 4 , 1 2 3 4 , 1 3 2 4 , 1 4 2 3 , 1 2 3 4 , 1 2 4 3 , and 1 2 3 4 . The r e fore the number of sets of tr ee patterns with 4 leaves is e qual to 2 15 = 327 6 8 , so a complete classification of W ilf classes is already a very heavy task. W e shall pr esent a very simple result on classification on W ilf classes for sets consisting of a single pattern. P A TTERN A VOIDANCE IN LABEL LED TREES 17 Theorem 6. There e x ist exac tly five W ilf classes of se ts of one p attern with four l eaves. Proo f. This theorem follows from a seq uence of lemmas which give a pr e cise description of the five W ilf classes. Lemma 6.1. The three patterns 4 3 1 2 , 2 3 1 4 , and 1 2 3 4 are Wilf equivalent to each other . Proo f. There is a bijective proof which is completely a nalogous to that of Lemma 5.1; we leave it to the r eader to fill in the details.  Lemma 6.2. The six patterns 1 2 3 4 , 1 3 2 4 , 1 4 2 3 , 2 1 3 4 , 1 2 3 4 , and 1 2 4 3 are Wilf equivalent to each other . Proo f. Let us denote P 1 = 1 2 3 4 , P 2 = 1 3 2 4 , P 3 = 1 4 2 3 , P 4 = 2 1 3 4 , P 5 = 1 2 3 4 , a nd P 6 = 1 2 4 3 . Let us show that P 1 ∼ W P 2 by e xhibiting a one-to-one corr espon- dence α between the P 1 -avoiding patterns and the P 2 -avoiding ones. If a tree T avoids both P 1 and P 2 , we put α ( T ) = T . If T avoids P 1 but contains P 2 , we may assume that there is an occurr e nce of P 2 at the r oot of T (otherwise we find the internal vertices closes t to the root that are root s of occurrences of P 2 , and apply α recur sively at these vertices). Let T = T 1 T 2 T 3 T 4 , and denote S i = α ( T i ), i = 1 , . . . , 4. W e put α ( T ) = S 1 S 3 S 2 S 4 . W e construc ted a bijection between P 1 -avoiding tr ees containing P 2 and P 2 -avoiding trees containing P 1 . The case of P 1 and P 3 is hand le d in a similar way . Let us show that P 1 ∼ W P 4 by e xhibiting a one-to-one corr espon- dence β between the P 1 -avoiding p atterns and the P 4 -avoiding ones. If a tr ee T avoids both P 1 and P 4 , we put β ( T ) = T . I f T a voids P 1 but contains P 4 , we may assume that there is an occurr ence of P 4 at the root of T (otherwise we find the internal vertices c losest to the roo t that are roo ts of occurr ences of P 4 , and apply β recursively at these vertices). Let T = T 2 T 1 T 3 T 4 , and denote S 1 = β ( T 1 T 2 ), S 2 = β ( T 3 T 4 ). W e put β ( T ) = S 1 S 2 . Note that the only vertex of this tr ee where an 18 VLADIMIR DOTSENKO occurr ence of P 4 can be r ooted is the root. However , if there is an occurr ence of P 4 ther e , it is easily see n to imply an occurrence of P 1 in T , a contradiction. W e construct e d a bijection between P 1 -avoiding tr ee s containing P 2 and P 2 -avoiding trees containing P 1 . The equivalence P 5 ∼ W P 6 can be established inductively similarly to how it is done in Lemma 5.1. Finally , the easiest way to see the equivalence P 1 ∼ W P 5 is via the inverse generating functions. Basically , P 1 and P 5 have the same stru ctur e of self-overlaps: there a re two self-overlaps, one of which is “rigid” (only one labe lling of lea ves is consistent with the local incr e a sing condition), and the other one admits three di ff erent leaf labellings. This allows for an inductively constructed bijections be- tween the clusters that control the coe ffi cients of the in verse series. W e leave the details to the reader .  Lemma 6.3. The four patterns 3 4 1 2 , 4 2 1 3 , 2 4 1 3 , and 3 2 1 4 are Wilf equivalent to each other . Proo f. The corresponding permutation patterns are W ilf equivale nt, so the Corollary 3 applies.  Combining the lemmas above with a somewhat lengthy computa- tion showing that • for the class described in Lem ma 6.1 the sequence count- ing the tr ees avoiding the patterns of that class begins with 1 , 1 , 3 , 14 , 91 , 756 , 765 7, • for the class described in Lem ma 6.2 the sequence count- ing the tr ees avoiding the patterns of that class begins with 1 , 1 , 3 , 14 , 90 , 739 , 739 2, • for the class described in Lem ma 6.3 the sequence count- ing the tr ees avoiding the patterns of that class begins with 1 , 1 , 3 , 14 , 90 , 738 , 736 4, • for the pattern 3 1 2 4 the sequence counting the tr ee s a voiding that pattern begins with 1 , 1 , 3 , 1 4 , 90 , 740 , 742 0, • for the pattern 4 1 2 3 the sequence counting the tr ee s a voiding that pattern begins with 1 , 1 , 3 , 1 4 , 90 , 737 , 733 6, we conclude that there are exactly five W ilf classes.  Of the five integer seque nces we saw in the previous pro of, only two seem to appear in the Online Encyclopedia of Integer Sequences [38]: the thir d one matches A088789 , the seque nce of coe ffi cients in P A TTERN A VOIDANCE IN LABEL LED TREES 19 the compositional inverse of the power series 2 x 1 + exp( x ) , and the first one matches A183611 , which, if we take care of the di ff erence in number- ing, is described as the sequence of coe ffi cients of t he power series f ( z ) satisfying the di ff er ential equation f ′′ ( z ) = f ′ ( z ) 2 + z f ′ ( z ) 3 . The first of these descriptions is not too surprising, as the inversion formula (34) suggests that the inverse series ma kes lots of sense combinato- rially . The second description is related to the results of Khor oshkin and Piontkovski [22] who proved that in some cases the generating function does indeed satisfy a di ff erential equation; however , the patterns of that W ilf class are not covered by their result s, so the appearance of a di ff erential eq ua tion in this enumeration prob lem would be another bit of evidence supporting their gene ral conjec- tur e that states that for every finite set of patterns the corresponding generating function satisfies a non-linear di ff erential equation with polynomial coe ffi cients. It is natural to ask which tr e e patterns are “the hardest to avoid”, that ha ve the fewest numbers of trees that a void them, and which tr ee patterns are “the easiest to avoid”, that is have the largest numbers of trees that avoid them. After e xamining the pr oof of the previous theor em and performing some computer experiments, we arrived at the following conjectur e which is closely related to the conjectur e of Elizalde and Noy [12] that the permutation 12 . . . n (the identity permutation) is the easiest to avoid, and to the conjectur e of Na ka- mura [31] that the permutation 12 . . . ( n − 2) n ( n − 1) (the transposition of its last two entries) is the hardest to avoid. Conjecture 2. Let us denote by LC < n , LC > n , and RC n the left c omb with n leaves whose leaf labels increase along the w ay from the root , the left comb with n leaves wh ose leaf labels decrease along the way from the root, and the right comb with n leaves respectivel y . The patterns LC < n , LC > n , and RC n are the easiest to avoid, and the p attern RC n − 1 n is the hardes t to avoid. 5.2. Case X = X 2 , | X | = 2 . Thro ughout this section we assume that we work with binary tr ees with two possible labels for internal ver- tices, in other wor ds, X = X 2 = {◦ , •} . The following result is still e asy to obtain “by hand”. Theorem 7. There exis t exactly two Wilf class es of sets of one p attern with three leaves , and exactl y ten Wilf c lasses of sets of two patterns wi th three leaves. Proo f. First of all, let us note that because of the inversion formula (35) , we can work with the tr ees avoiding 11 and 10 patterns respectively . For the avoidance of 11 pa tterns, the only allowed pattern can either use only one internal vertex label (in which case Theor e m 5 mea ns that ther e exists only one allowed tr ee for each number of lea ves) or 20 VLADIMIR DOTSENKO use two di ff erent internal vertex labels (in which case there is no way to build an allowed tr ee with 4 or mor e leaves). For the avoidance of 10 pa tterns, our theor e m follows fr om a se- quence of lemmas which give a precise description of the ten W ilf classes. Lemmas 7.1–7.7 ar e almost obvious, since the trees that are being counted admit very explicit descriptions; we omit their pr oofs. Note that we switch to complements again: instead of listing the ten- element sets, we list their complements in the set of all tree patterns with three leaves. Lemma 7.1. The six sets whose comp lements in the set of all tree pat- terns wi th three leaves are { • • 3 1 2 , • • 2 1 3 } , { • • 2 1 3 , 3 • 2 • 1 } , { • • 3 1 2 , 3 • 2 • 1 } , { ◦ ◦ 3 1 2 , ◦ ◦ 2 1 3 } , { ◦ ◦ 2 1 3 , 3 ◦ 2 ◦ 1 } , { ◦ ◦ 3 1 2 , 3 ◦ 2 ◦ 1 } ar e Wilf equivalent to eac h other; the number of trees with n leaves avoiding either of these sets is equal to ( n − 1)! for each n ≥ 4 . Lemma 7.2. The two sets whose c om plements in the set of all tree patterns with three leaves are { • ◦ 3 1 2 , 3 ◦ 2 • 1 } , { ◦ • 3 1 2 , 3 • 2 ◦ 1 } are Wilf equivalent to each other; the number of trees with n leaves avoid i ng ei the r of thes e sets is equal to 3 2  n 2  ! for even n ≥ 4 and to  n − 1 2  ! + 1 2  n + 1 2  ! for odd n ≥ 3 . Lemma 7.3. Th e ten sets whose complem ents in the set of all tree pat- terns wi th three leaves are { • • 3 1 2 , • ◦ 2 1 3 } , { • • 2 1 3 , • ◦ 3 1 2 } , { • • 3 1 2 , ◦ • 2 1 3 } , { • • 2 1 3 , ◦ • 3 1 2 } , { ◦ ◦ 3 1 2 , • ◦ 2 1 3 } , { ◦ ◦ 2 1 3 , • ◦ 3 1 2 } , { ◦ ◦ 3 1 2 , ◦ • 2 1 3 } , { ◦ ◦ 2 1 3 , ◦ • 3 1 2 } , { 3 • 2 • 1 , ◦ • 3 1 2 } , { 3 ◦ 2 ◦ 1 , • ◦ 3 1 2 } are Wilf e quivale nt to each other; the number of trees with n leaves avoiding either of these sets is equal to n − 1 for each n ≥ 4 . Lemma 7.4. The 32 sets whose compl ements in the set of all tree pat- terns wi th three leaves are { • • 3 1 2 , ◦ ◦ 3 1 2 } , { • • 3 1 2 , ◦ ◦ 2 1 3 } , { • • 3 1 2 , 3 ◦ 2 ◦ 1 } , { • • 2 1 3 , ◦ ◦ 3 1 2 } , { • • 2 1 3 , ◦ ◦ 2 1 3 } , { • • 2 1 3 , 3 ◦ 2 ◦ 1 } , { 3 • 2 • 1 , ◦ ◦ 3 1 2 } , { 3 • 2 • 1 , ◦ ◦ 2 1 3 } , { 3 • 2 • 1 , 3 ◦ 2 ◦ 1 } , { • • 3 1 2 , ◦ • 3 1 2 } , { • • 2 1 3 , ◦ • 2 1 3 } , { ◦ ◦ 3 1 2 , • ◦ 3 1 2 } , { ◦ ◦ 2 1 3 , • ◦ 2 1 3 } , { 3 • 2 • 1 , 3 ◦ 2 • 1 } , { 3 ◦ 2 ◦ 1 , 3 ◦ 2 • 1 } , { 3 • 2 • 1 , 3 • 2 ◦ 1 } , { • • 2 1 3 , • ◦ 2 1 3 } , { ◦ ◦ 3 1 2 , ◦ • 3 1 2 } , P A TTERN A VOIDANCE IN LABEL LED TREES 21 { ◦ ◦ 2 1 3 , ◦ • 2 1 3 } , { 3 • 2 • 1 , 3 ◦ 2 • 1 } , { 3 ◦ 2 ◦ 1 , 3 • 2 ◦ 1 } , { • ◦ 3 1 2 , ◦ • 3 1 2 } , { • ◦ 2 1 3 , ◦ • 2 1 3 } , { 3 • 2 ◦ 1 , 3 ◦ 2 • 1 } , { 3 • 2 • 1 , ◦ • 2 1 3 } , { 3 ◦ 2 ◦ 1 , • ◦ 2 1 3 } , { 3 ◦ 2 • 1 , • • 3 1 2 } , { 3 ◦ 2 • 1 , • • 2 1 3 } , { ◦ ◦ 3 1 2 , 3 • 2 ◦ 1 } , ◦ ◦ 2 1 3 , 3 • 2 ◦ 1 } , { • ◦ 2 1 3 , 3 ◦ 2 • 1 } , { ◦ • 2 1 3 , 3 • 2 ◦ 1 } are Wilf equiva- lent to each other; the number of trees with n leaves avoiding e ither of thes e sets is equal to 2 for each n ≥ 4 ,. Lemma 7.5. The two sets whose c om plements in the set of all tree patterns with three leaves are { ◦ • 3 1 2 , 3 ◦ 2 • 1 } , { • ◦ 3 1 2 , 3 • 2 ◦ 1 } are Wilf equivalent to each other; the number of trees with n leaves avoid i ng ei the r of thes e sets is equal to 1 for n = 4 , and equal to 0 for each n ≥ 5 . Lemma 7.6. The two sets whose c om plements in the set of all tree patterns with three leaves are { ◦ • 2 1 3 , 3 ◦ 2 • 1 } , { • ◦ 2 1 3 , 3 • 2 ◦ 1 } are Wilf equivalent to each other; the number of trees with n leaves avoid i ng ei the r of thes e sets is equal to 2 for n = 4 , and equal to 0 for each n ≥ 5 . Lemma 7.7. The two sets whose c om plements in the set of all tree patterns with three leaves are { ◦ • 3 1 2 , ◦ • 2 1 3 } , { • ◦ 3 1 2 , • ◦ 2 1 3 } are Wilf equivalent to each other; the number of trees with n leaves avoid i ng ei the r of thes e sets is equal to 0 for each n ≥ 4 . Lemma 7.8. The six sets whose comp lements in the set of all tree pat- terns wi th three leaves are { • • 3 1 2 , 3 • 2 ◦ 1 } , { • • 2 1 3 , 3 • 2 ◦ 1 } , { ◦ ◦ 3 1 2 , 3 ◦ 2 • 1 } , { ◦ ◦ 2 1 3 , 3 ◦ 2 • 1 } , { 3 • 2 • 1 , • ◦ 2 1 3 } , { 3 ◦ 2 ◦ 1 , ◦ • 2 1 3 } ar e Wilf equivalent to eac h other; the number of trees with n leaves avoiding either of these sets is equal to the number of involutions in S n − 1 for each n ≥ 4 . Proo f. Examining the structur e of the allowed trees, we see that in each case we have a right comb or an incr e asing (decr easing) left comb, possibly with le a ves replaced by pairs of le aves. Examining the leaf labels, one easily extract a decomposition of an involut ion into the pr oduct of disjoint cycles, which gives a one-to-one correspondence.  Lemma 7.9. The two sets whose c om plements in the set of all tree patterns with three leaves are { 3 • 2 • 1 , • ◦ 3 1 2 } and { 3 ◦ 2 ◦ 1 , ◦ • 3 1 2 } are Wilf equivalent 22 VLADIMIR DOTSENKO to each other; the number of trees with n leaves avoiding either of these sets is equal to the n th Fibonacci number ( f 0 = 0 , f 1 = 1 , f n + 1 = f n + f n − 1 ) for each n ≥ 3 . Proo f. Examining the structur e of the allowed trees, we see that in each case we have a right comb, possibly with leaves r eplaced by pairs of leaves, but with the leaf la bels incr e asing globally along the path from the root, ther e fore the number of tr ee s we are trying to compute is equa l to the numbe r of sequences of 1’s and 2’s that sum up to n − 1, which is known to be the Fibonacci number [39].  Lemma 7 . 1 0. Th e two sets whose c omplements in the set of all tr ee patterns with three leaves are { • ◦ 2 1 3 , ◦ • 3 1 2 } , { ◦ • 2 1 3 , • ◦ 3 1 2 } are Wilf equivalent to each other; the number of trees with n leaves avoid i ng ei the r of thes e sets is twice the number of alternating per mutations in S n − 1 for each n ≥ 3 . Proo f. Clearly , each allowed tr e e is a left comb, the labels ◦ , • along the path fr om the roo t alternate, and so do the le af labels. The r e fore, the statement is obvious: alternating permutations come fr om the leaf labels, and “twice” reflect the fact that for each k there are two alternating seq uences of ◦ ’s and • ’s of length k .   Further (computer-aided) investigation shows that the following statement is true: Theorem 8. For sets of patterns with 3 leave s, there are (1) 2 Wilf classe s of 1 -e l ement sets, (2) 10 Wilf class e s of 2 -element sets, (3) 40 Wilf class e s of 3 -element sets, (4) 99 Wilf class e s of 4 -element sets, (5) 189 Wilf classes of 5 -element s e ts, (6) 202 Wilf classes of 6 -element s e ts, (7) 189 Wilf classes of 7 -element s e ts, (8) 99 Wilf class e s of 8 -element sets, (9) 40 Wilf class e s of 9 -element sets, (10) 10 Wilf classes of 10 -el e ment sets, (11) 2 Wilf classes of 11 -ele ment sets. T o conclude this section, l e t us mention a promising direction towar ds ne w bijective proofs for enumeration of W ilf equivalence classes. In the case of unlabelled planar trees, ma ny bijections have been constructed in [13], where the “word notation” for tr ee s was used. That “word notation” is a constr uction of crucial importance for the operad theory . It was discover e d by Ho ff be ck [18] who de- fined a partial ordering compatible with the operad structur e on P A TTERN A VOIDANCE IN LABEL LED TREES 23 the set of “tree monomials” which was then used to formulate and pr ove a PBW criterion for Koszul operads. Later , this partial or- dering was extended to a total ordering (still compatible with the operad struc tur e) in [8]; that or de ring is one of the key ingredients of the Gr ¨ obner bases machinery in the case of operads. The latter total ordering is coming from replacing a tree T ∈ L T ( X ) with a pair ( path ( T ) , per m ( T )) consisting of a “path seq uence” path ( T ) (a se- quence of words in the alphabet X ) and a permutation per m ( T ) of the set of le a ves of T . All questions of pattern avoidance for trees can be translated into questions of pattern a voidance for this type of data; the corr esponding notion of a pattern is a certain mixture of the classical notion of a divisor of a word and the notion of a generalised pattern in permutations [2, 3] (di ff erent from the na¨ ıve notion of a generalised pattern in coloured permutations). This way of thinking about patterns in trees should be useful for bijective proofs ; we hope to addr ess it in mor e detail e lsewhe r e . A ppendix A . S huffle operads and p a tterns in trees In this appendix, we recall some r e levant d e finitions of the operad theory , and explain how the notion of pattern avoidance in trees arises naturally in this context. Let us denote by Ord the category whose objects are non-empty finite order ed sets (with order -preserv ing bijections as morphisms). Also, we denote by V ect the category of vector spaces (with lin- ear operators as morphisms). It is usually e nough to assume vec- tor spaces finite-dimensional, though sometimes more generality is needed, a nd one assumes, for instance, that they ar e graded with finite-dimensional homogeneous components. Definition 5. (1) A (non-symme tr i c) colle ction is a contravariant functor from the category Ord to the category V ect. W e shall r efer to images of individual sets as components of our collec- tion. (2) Let P and Q be two non-symmetric collections. The shu ffl e composition product of P and Q is the non-symmetric collec- tion P ◦ sh Q defined by the formula ( P ◦ sh Q )( I ) : = M k P ( k ) ⊗         M f : I ։ [ k ] Q ( f − 1 (1)) ⊗ . . . ⊗ Q ( f − 1 ( k ))         , wher e the sum is taken over all sh u ffl ing surjections f , that is surjections for which min f − 1 ( i ) < min f − 1 ( j ) whenever i < j . (3) A shu ffl e operad is a monoid in the category of non-symmetric collections equipped with the shu ffl e composition p roduct. 24 VLADIMIR DOTSENKO This definition is a counterpart of a more classical one, where one deals with finite sets without a specified order , and all surjections ar e allowed to de fine the composition. The corr esponding monoids ar e called (symmetric) operads, and those are monoids widely used in algebra and topology , since they capture the algebraic properties of compositions of operations with several arg uments. In particular , algebras of a certain type, like all associative algebras, or all Lie algebras, can be viewe d as modules over a monoid of this kind (formed by all operations that can be d e fined on algebras of the given type), and this point of view proves to be useful. However , for computational purposes the fact that finite sets have symmetries gets in the way , and it turns out that shu ffl e operads allow to deal with some of the troubles arising because of that. (Every symmetric operad can be viewed as a shu ffl e operad, since every order ed set can be viewed as an unor dered set.) Monoids in the monoidal categor y of vector spaces (with the usual tensor product) are associative algebras. One can p resent a ssociative algebras via generators and relations; if all relations are monomials in gene rators, the corr esponding algebra admits a straightforwar d basis consisting of all monomials avoiding the monomials from the set of relations. In general, there is no such elegant description; to obtain it, one uses the machinery of Gr ¨ obner bases. A Gr ¨ obner basis is a special choice of a se t of relations that allows to find a “monomial r eplacement” for the given algebra A , that is an algebra with the same generators an d with monomial relations for which the natural monomial basis is also a basis for A . Such a set of monomial r elations is pro vided by the “leading terms” of the r elations forming a Gr ¨ obner basis. It is very na tural to try a nd find an a p pr opriate Gr ¨ obner bases theory for operads. For operads with symmetries, it turns out to be impossible: not every symmetric operad has a monomial replace- ment. However , for shu ffl e operads it turns out to be possible. Let us be a little bit more precise. Similarly to the case of associative alge- bras, a shu ffl e operad can be presented via generators and r elations, that is as a quotient of the free operad F ( V ), where V is the space of generators (which itself is a non-symmetric collection). If X is the collection of or dered bases for components of V , that is a functor fr om Or d to Ord, then the free shu ffl e operad generated by V adm its a basis of “ tree monomials” which can be defined combinatorially; they are precisely planar X -labelled r ooted trees studied thr oughout this paper . Any shu ffl e composition of tr ee monomials is again a tr ee monomial. The crucial feature of shu ffl e operads is that they admit good monomial or derings (and therefor e one can talk about leading terms of r elations). Mor e precisely , there exist several wa ys to introduce a total ordering of tr ee monomials in such a way that all P A TTERN A VOIDANCE IN LABEL LED TREES 25 the shu ffl e compositions respect that total or de ring: increasing one of the monomials we compose incr eases the r e sult. Furthermor e, the algebraic statement that a tree monomial T is obtained from another tr ee monomial S by shu ffl e compositions (that is, T is divisible by S in our monoid) means, in the combinatorial language, that S occurs in T as a pattern. W e now have all the ingredients to relate questions about shu ffl e operads to pa ttern avoidance in trees. A Gr ¨ obner basis of an ide al I in the free shu ffl e operad is a system G of generators of I for which the leading monomial of every elem e nt of I is divisible by one of the leading terms of eleme nts of G . Such a system of gene rators allows to perform “long division” modulo I , computing for every element its canonical repr esentative. T ree monomials avoiding the leading terms of elements of G (“normal monomials”) form a basis in the quotient by the ideal I ; in other wor ds, shu ffl e operads do admit monomial r eplacements. 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