Wilf-equivalence on k-ary words, compositions, and parking functions

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1 Wilf-equivalene on k-ary ords, ompositions, and parking funtions Vít Jelínek Department of Applied Mathematis, Charles University, Prague Toufik Mansour Department of Mathematis, Haifa University, Haifa, Israel Submitted: May 9, 2008; Aepted: May 4, 2009; Published: May 11, 2009 Mathematis Subjet Classifiation: Primary 05A18; Seondary 05E10, 05A17, 05A19 Abstrat In this paper, e study pattern-avoidane in the set of ords over the alphabet [k]. We say that a ord [k] n ontains a pattern τ [l] m, if ontains a subsequene order-isomorphi to τ. This notion generalizes pattern-avoidane in permutations. We determine all the Wilf-equivalene lasses of ord patterns of length at most six. We also onsider analogous problems ithin the set of integer ompositions and the set of parking funtions, hih may both be regarded as speial types of ords, and hih ontain all permutations. In both these restrited settings, e determine the equivalene lasses of all patterns of length at most five. As it turns out, the full lassifiation of these short patterns an be obtained ith only a fe general bijetive arguments, hih are appliable to patterns of arbitrary size. 1 Introdution In this paper, e study patterns avoidane in the domains of ords, integer partitions, and parking funtions. This an be seen as an extension of the frequently studied onept of pattern avoidane of permutation patterns. Our results extend previous ork of Burstein [4], ho desribed the equivalene lasses of k-ary ords of length at most 3, and Supported by the projet MSM of the Czeh Ministry of Eduation, and by the grant GD201/05/H014 of the Czeh Siene Foundation. the eletroni journal of ombinatoris 16 (2009), #R58 1

2 of Savage and Wilf [14], ho desribed the equivalene lasses of integer ompositions of length at most 3. Our lassifiation is largely based on several ne bijetive arguments, inspired by the ideas from Krattenthaler [11], Bakelin, West and Xin [3], and Jelínek and Mansour [9]. 1.1 k-ary ords Let [k] = {1, 2,..., k} be a totally ordered alphabet of k letters, and let [k] n denote the set of ords of length n over this alphabet. Consider to ords, σ [k] n and τ [l] m. Assume additionally that τ ontains all letters 1 through l (a ord ith this property ill be alled a pattern). We say that σ ontains an ourrene of τ, or simply that σ ontains τ, if σ has a subsequene orderisomorphi to τ, i.e., if there exist 1 i 1 <... < i m n suh that, for any to indies 1 a, b m, σ ia < σ ib if and only if τ a < τ b. If σ ontains no ourrenes of τ, e say that σ avoids τ. For a pattern τ, let [k] n (τ) denote the set of k-ary ords of length n hih avoid the pattern τ. Let f τ (n, k) be the number of τ-avoiding ords in [k] n, i.e., f τ (n, k) = [k] n (τ). We say that to patterns τ and τ are ord-equivalent (or, more briefly, -equivalent), and e rite τ τ, if for all values of k and n, e have f τ (n, k) = f τ (n, k). There are to operations on ords hih trivially preserve the -equivalene, alled the reversal and the omplement. The reversal of a ord τ [k] m, denoted by r(τ), is obtained by riting the letters of τ in the reverse order, i.e., the i-th letter of r(τ) is equal to the (m i+1)-th letter of τ. The omplement of a ord τ, denoted by (τ), is obtained by turning τ upside-don, i.e., a letter j is replaed by the letter l j+1, here l is the largest letter of τ. For example, if l = 3, m = 4, then r(1232) = 2321, (1232) = 3212, r((1232)) = (r(1232)) = Clearly, r = r and r 2 = 2 = ( r) 2 = id, so r, is a group of symmetries of a retangle. Several authors have previously onsidered pattern avoidane in ords [1, 2, 4, 5, 10, 13]. In 1998, Burstein [4] proved that In 2002, Burstein and Mansour [5] proved that By these to results e obtain that there are 3 -equivalene lasses of patterns of length three: 111, , Compositions A omposition σ = σ 1 σ 2... σ m of n N is an ordered olletion of one or more positive integers hose sum is n. The numbers σ 1,...,σ m are alled parts of the omposition. We let C n denote set of all ompositions of n. We say that the omposition σ C n ontains a pattern τ [l] s, if σ ontains a subsequene order-isomorphi to τ. Let C n (τ) denote the set of all the ompositions in the eletroni journal of ombinatoris 16 (2009), #R58 2

3 C n that avoid τ. We say that to patterns τ and τ are omposition-equivalent (or just -equivalent), and e rite τ τ, if for all values of n, e have C n (τ) = C n (τ ). It is easy to see that every pattern is -equivalent to its reversal. Hoever, a pattern does not need to be -equivalent to its omplement. Savage and Wilf [14] onsidered pattern avoidane in ompositions for a single pattern τ S 3 (S 3 is the set of the permutations on three letters), and shoed that the number of ompositions of n avoiding τ S 3 is independent of τ, that is, Reently, Heubah, Mansour and Munagi [7] shoed that and These to results omplete the lassifiation of patterns of length three in ompositions, and sho they form exatly 4 -equivalene lasses: , , , Parking funtions A ord σ [k] n is alled a parking funtion if for every i = 1,..., n, σ has at least i letters smaller or equal to i. Let PF n denote the set of parking funtions of length n, and let PF n (τ) be the set of the parking funtions of length n that avoid a pattern τ. We say that to patterns τ, τ are p-equivalent, denoted by τ p τ, if for every n, the to sets PF n (τ) and PF n (τ ) have the same ardinality. Clearly, eah pattern is p-equivalent to its reversal. Although some enumerative aspets of parking funtions have been previously studied [6, 12, 15], e are not aare of any previous results dealing ith pattern avoidane in this setting. 1.4 Strong equivalene of ords We no introdue an equivalene relation on ords, hih refines all the equivalenes mentioned above. For a ord σ of length n, the ontent of σ is the unordered multiset of the n letters appearing in σ. In partiular, to ords have the same ontent, if one an be obtained from the other by a suitable rearrangement of letters. We say that to patterns τ, τ are strongly equivalent, denoted by τ s τ, if for every k, n there is a bijetion f beteen [k] n (τ) and [k] n (τ ) ith the property that for every σ [k] n (τ), the ord f(σ) has the same ontent as σ. Clearly, if to patterns are strongly equivalent, then they are also -equivalent, -equivalent and p-equivalent. Eah pattern is strongly equivalent to its reversal, and if to patterns τ and τ are strongly equivalent, then their omplements (τ) and (τ ) are strongly equivalent as ell. In this note, e adapt previous results on fillings of diagrams [11], as ell as results on pattern-avoidane in set partitions [9] to desribe several types of ontent-preserving the eletroni journal of ombinatoris 16 (2009), #R58 3

4 bijetions beteen pattern-avoiding families of ords. Using systemati omputer enumeration of small patterns, e verify that these bijetions, together ith the reversal and omplement operations, are suffiient to desribe all the -equivalent patterns of length at most six. Similarly, e verify that our results on strong equivalene desribe all the -equivalene and p-equivalene lasses of patterns of size at most five. In partiular, for patterns of size at most five, -equivalene lasses oinide ith p-equivalene lasses (but not ith -equivalene lasses, beause -equivalene is losed under omplementation and reversal, hereas p- and -equivalene is only losed under reversal). In the appendix, e briefly summarize the equivalene lasses of small patterns, ith respet to -, -, and p-equivalene. The full enumeration data and the soure odes of the omputer programs e used are available on the ebsite of the seond author [16, 17, 18, 19, 20, 21]. 2 Strongly equivalent families In this setion, e use several tehniques previously applied in the ontext of fillings of Ferrers diagrams to obtain lasses of strongly equivalent ords. We may represent k-ary ords of length n as 0 1 matries ith k ros and n olumns and exatly one 1-ell in eah olumn. We assume that the ros of a matrix are numbered bottom-to-top, and the olumns are numbered left-to-right. For a ord σ of length n over the alphabet [k], let M(σ, k) be the k n matrix ith a 1-ell in ro i and olumn j if and only the j-th letter of σ is equal to i. With this representation, e may use knon bijetions on fillings of diagrams to obtain diretly ne equivalenes among ords. A Ferrers diagram is an array of ells hose olumns have noninreasing length, and the bottom ells of the olumns appear in the same ro. A filling of a Ferrers diagram is an assignment of zeros and ones into its ells suh that every olumn has exatly one 1-ell. We say that a filling of a Ferrers shape F ontains a matrix M if F has a (not neessarily ontiguous) retangular subshape hih indues a filling idential to M. We ill say that to matries M and M are Ferrersequivalent if for every Ferrers shape F the number of M-avoiding fillings is equal to the number of M -avoiding fillings. We say that M and M are strongly Ferrers-equivalent if for every Ferrers shape F there is a bijetion beteen M-avoiding and M -avoiding fillings of F that preserves the number of 1-ells in eah ro. The folloing lemma allos us to translate results about fillings of Ferrers shapes into results about ords. The lemma is based on an idea hih is often applied in the ontext of pattern-avoiding permutations [3], graphs [8] or set partitions [9]. For a ord ρ [l] n and an integer k, e let ρ + k denote the ord obtained by inreasing eah letter of ρ by k. Lemma 2.1. Let τ and τ be to patterns ith k letters, let ρ be a pattern ith l letters. If M(τ, k) and M(τ, k) are strongly Ferrers-equivalent then the to (k +l)-letter patterns τ(ρ + k) and τ (ρ + k) are strongly equivalent. (Here τ(ρ + k) denotes the onatenation of τ and ρ + k.) the eletroni journal of ombinatoris 16 (2009), #R58 4

5 Proof. Let us rite σ = τ(ρ + k) and σ = τ (ρ + k). For a given m and n, hoose a ord x [m] n (σ), and let M = M(x, m) be its orresponding matrix. Note that M avoids the matrix M(σ, k + l). Color the ells of M red and green, here a ell is green if and only if the submatrix of M stritly to the right and stritly to the top of ontains M(ρ, l), otherise the ell is red. Note that the green ells form a Ferrers diagram and that the nonzero olumns of this diagram indue an M(τ, k)-avoiding filling. Using the strong Ferrers-equivalene of M(τ, k) and M(τ, k), e may transform this filling into a M(τ, k)-avoiding filing. This operation transforms M into a matrix M representing a σ -avoiding ord x ith the same ontent as x. To see that this operation an be inverted, observe that the operation has only modified the filling of the green ells of M. Observe also that for every green ell of M, there is a opy of M(ρ, l) stritly to the right and stritly above hih only onsists of red ells. Thus the red ells of M oinide ith the red ells of M. We thus have a bijetion shoing that σ s σ. Using knon results about Ferrers equivalene [8, 9, 11], e obtain the folloing equivalenes, valid for any pattern ρ. Fat 2.2. M(12 k, k) is strongly Ferrers equivalent to M(k(k 1) 1, k) [11]. This implies that 12 k(ρ + k) s k(k 1) 1(ρ + k). Fat 2.3. M(2 i 12 j, 2) is strongly Ferrers-equivalent to M(12 i+j, 2), for any i, j 0 [9, Lemma 39]. This implies that 2 i 12 j (ρ + 2) s 12 i+j (ρ + 2). The above-mentioned results do not aount for all the equivalenes among ordpatterns of small length. To omplete our lassifiation, e need another lemma, hose proof uses an idea that has been previously applied in the ontext of pattern-avoiding set partitions [9, Theorem 48]. Lemma 2.4. For any k, all the patterns that onsist of a single symbol 1, a single symbol 3 and k 2 symbols 2 are strongly equivalent. Proof. Let k be fixed. Let τ(i, j) denote the ord of length k hose i-th symbol is 1, the j-th symbol is 3 and the remaining symbols are equal to 2. Our aim is to sho that all the patterns in the set {τ(i, j), i j, 1 i, j k} are strongly equivalent. Sine eah ord is strongly equivalent to its reversal, e only need to deal ith the ords τ(i, j) ith i < j. From Fat 2.3, e dedue that the ords {τ(1, j), j = 2,...,k} are all strongly equivalent, and the ords {τ(i, k), i = 1,..., k 1} are all strongly equivalent as ell. To prove the lemma, it suffies to sho that for every i < j < k, the ord τ(i, j) is strongly equivalent to the ord τ(i + 1, j + 1). Let m be an integer. We ill say that a ord σ ontains τ(i, j) at level m if there is a pair of symbols l, h suh that l < m < h, and suh that the ord σ ontains a subord over the alphabet {l, m, h} hih is orderisomorphi to τ(i, j). For example, the ord ontains the pattern 1223 at level 3 (due to the subord 1334), hile it avoids 1223 at level 2. the eletroni journal of ombinatoris 16 (2009), #R58 5

6 Assume no that e are given a fixed pair of indies i, j, ith i < j < k, and e ant to provide a ontent-preserving bijetion beteen τ(i, j)-avoiding and τ(i+1, j +1)- avoiding ords of length n. We ill say that a ord σ is an m-hybrid if for every m < m, the ord σ avoids τ(i, j) at level m, hile for every m m, σ avoids τ(i+1, j+1) at level m. We ill present, for any m 1, a ontent-preserving bijetion beteen m-hybrids and (m + 1)-hybrids. By omposing these bijetions, e obtain the required bijetion beteen τ(i, j)-avoiding and τ(i + 1, j + 1)-avoiding ords. Let m 1 be fixed. Let σ be an arbitrary ord. A letter of σ is alled lo if it is smaller than m, and a letter is alled high if it is greater than m. A lo luster of σ is a maximal blok of onseutive lo symbols of σ. A high luster is defined analogously. Thus, every symbol of σ different from m belongs to a unique luster. The landsape of σ is a ord over the alphabet {L, m, H} obtained by replaing every lo luster of σ by a single symbol L, and every high luster of σ by a single symbol H. Note that σ ontains τ(i, j) at level m if and only if the landsape of σ ontains the subsequene m i 1 Lm j i 1 Hm k j. We ill no desribe the bijetion beteen m-hybrids and (m + 1)-hybrids. Let σ be an m-hybrid ord, let X be its landsape. We split X into three parts X = P ms, here P is the prefix of X formed by all the symbols of X that appear before the first ourrene of m in X, and S is the suffix of all the symbols that appear after the first ourrene of m. Let us define a ord X by X = SmP. Note that X ontains a subsequene m i 1 Lm j i 1 Hm k j if and only if X ontains a subsequene m i Lm j i 1 Hm k j 1. Thus, sine X is a landsape of a ord that avoids τ(i + 1, j + 1) at level m, e kno that any ord ith landsape X must avoid τ(i, j) at level m. Let us define a ord σ by the folloing three rules. 1. The ord σ has landsape X. 2. For any p, the p-th lo luster of σ onsists of the same sequene of symbols as the p-th lo luster of σ. 3. For any q, the q-th high luster of σ onsists of the same sequene of symbols as the q-th high luster of σ. Clearly, there is a unique ord σ satisfying these properties. Note that the subsequene of all the lo symbols of σ is the same as the subsequene of all the lo symbols of σ, and these sequenes are partitioned into lo lusters in the same ay. An analogous property holds for the high symbols too. We laim that σ is an (m + 1)-hybrid. We have already pointed out that σ avoids τ(i, j) at level m. Let us no argue that σ avoids τ(i, j) at level m, for every m < m. For ontradition, assume that σ ontains a subsequene T = m i 1 lm j i 1 hm k j, for some l < m < h. If h < m, then all the symbols of T are lo, and sine σ has the same subsequene of lo symbols as σ, e kno that σ also ontains T as a subsequene, ontraditing the assumption that σ is an m-hybrid. Assume no that h m. Let x and y be the to symbols adjaent to h in the sequene T (note that h is not the last symbol of T, so x and y are ell defined). Both x and y are the eletroni journal of ombinatoris 16 (2009), #R58 6

7 lo, and they belong to distint lo lusters of σ, beause the symbol h is not lo. Sine the lo symbols of σ are the same as the lo symbols of σ, and they are partitioned into lusters in the same ay, e kno that σ ontains a subsequene m i 1 lm j i 1 h m k j, here h is a non-lo symbol. This shos that σ ontains τ(i, j) at level m, hih is impossible, beause σ is an m-hybrid. By an analogous argument, e may sho that σ avoids τ(i + 1, j + 1) at any level m > m. We onlude that the mapping desribed above transforms an m-hybrid σ into an (m + 1)-hybrid σ. It is lear that the mapping is reversible and provides the required bijetion beteen m-hybrids and (m + 1)-hybrids. Appendix A: the -equivalene lasses In Tables 1, 2 and 3, e list the nontrivial -equivalene lasses of patterns of size 4, 5 and 6, respetively. From eah symmetry lass (i.e., a lass generated by reversal and omplement of a single pattern) e only list the lexiographially minimal pattern. We only list the -equivalene lasses that have at least to nonsymmetri elements Table 1: -equivalene lasses of patterns of size Table 2: -equivalene lasses of patterns of size Table 3: -equivalene lasses of patterns of size 6 the eletroni journal of ombinatoris 16 (2009), #R58 7

8 Appendix B: p-equivalene and -equivalene lasses In Tables 4 and 5, e list the nontrivial -equivalene lasses for patterns of size 4 and 5, respetively. For patterns of these sizes, the -equivalene lasses oinide ith p- equivalene lasses. A symmetry lass of a pattern is generated by the reversal operation. We again onsider only one representative of eah symmetry lass, and e only list the -equivalene lasses ith at least to nonsymmetri elements Table 4: -equivalene lasses of patterns of size Table 5: -equivalene lasses of patterns of size 5 Referenes [1] M. Albert, R. Aldred, M.D. Atkinson, C. Handley, and D. Holton, Permutations of a multiset avoiding permutations of length 3, Europ. J. Combin. 22 (2001) [2] N. Alon and E. Friedgut, On the number of permutations avoiding a given pattern, J. Combin. Theory Series A 89 (2000) [3] J. Bakelin, J. West, and G. Xin, Wilf-equivalene for singleton lasses, Adv. Appl. Math. 32:2 (2007) [4] A. Burstein, Enumeration of ords ith forbidden patterns, Ph.D. thesis, University of Pennsylvania, [5] A. Burstein and T. Mansour, Words restrited by patterns ith at most 2 distint letters, Eletron. J. Combin. 9:2 (2002), #R3. [6] S.-P. Eu, T.-S. Fu, and C.-J. Lai, On the enumeration of parking funtions by leading terms, Adv. in Appl. Math. 35:4 (2005) [7] S. Heubah, T. Mansour, and A. Munagi, Avoiding permutation patterns of type (2, 1) in ompositions, preprint. [8] A. de Mier, k-nonrossing and k-nonnesting graphs and fillings of Ferrers diagrams, Eletroni Notes in Disrete Mathematis 28 (2007) the eletroni journal of ombinatoris 16 (2009), #R58 8

9 [9] V. Jelínek and T. Mansour, On pattern-avoiding partitions Elet. J. Combin. 15:1 (2008), #R39. [10] M. Klazar, The Füredi-Hajnal onjeture implies the Stanley-Wilf onjeture, Formal poer series and algebrai ombinatoris (Moso,2000), Springer, Berlin (2000) [11] C. Krattenthaler, Groth diagrams, and inreasing and dereasing hains in fillings of Ferrers shapes, Adv. Appl. Math. 37:3 (2006) [12] A. Rattan, Permutation fatorizations and prime parking funtions, Ann. Comb. 10:2 (2006) [13] A. Regev, Asymptotis of the number of k-ords ith an l-desent, Elet. J. Combin. 5 (1998), #R15. [14] C. D. Savage and H. S. Wilf, Pattern avoidane in ompositions and multiset permutations, Adv. Appl. Math. 36:2 (2006) [15] C. Zara, Parking funtions, stak-sortable permutations, and spaes of paths in the Johnson graph, Eletron. J. Combin. 9:2 (2002/03) #R11. [16] [17] [18] [19] ords.html [20] ompositions.html [21] parkings.html the eletroni journal of ombinatoris 16 (2009), #R58 9