In this paper, we compute and demonstrate the equivalence of the joint distribution of the first ... more In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the n-th large Schröder number and thus one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics, special generating functions (specific to each class) and the kernel method to establish our results. In some cases, we utilize the conjecture itself in a creative way to aid in solving the system of functional equations satisfied by the associated generating functions.
In this paper, we compute the distribution of the first letter statistic on nine avoidance classe... more In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schröder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.
ABSTRACTWe introduce Peano words, which are words corresponding to finite approximationsof the Pea... more ABSTRACTWe introduce Peano words, which are words corresponding to finite approximationsof the Peano space filling curve. We then find the number of occurrences of certainpatterns in these words.Keywords: Combinatorics on words, the Peano curve, generalized pattern 1. Introduction and BackgroundWe write permutations as words π = a 1 a 2 ···a n , whose letters are distinct and usuallyconsist of the integers 1,2,...,n.An occurrence of a pattern p in a permutation π is “classically” defined as a sub-sequence in π (of the same length as the length of p) whose letters are in the samerelative order as those in p. Formally speaking, for r ≤ n, we say that a permutationσ in the symmetric group S n has an occurrence of the pattern p ∈ S r if there exist1 ≤ i 1 < i 2 < ··· < i r ≤ n such that p = σ(i 1 )σ(i 2 )...σ(i r ) in reduced form. Thereduced form of a permutation σ on a set {j 1 ,j 2 ,...,j r }, where j 1 < j 2 < ··· < j r , isa permutation σ 1 obtained by renaming the le...
The problems of deciding the length of the longest increasing subsequence for a permutation or an... more The problems of deciding the length of the longest increasing subsequence for a permutation or an involution, the expectation and the limiting distribution, surveyed in Stanley’s ICM (International Congress of Mathematicians) Plenary address, have been widely studied by combinatorialists, analysts and probabilists. Partially motivated by the intriguing phenomenon stated by Simion and Schmidt [European J. Combin. 6 (1985) 383–406] that 231-avoiding permutations are exactly the set of layered permutations, in this paper, we investigate the limiting behavior of the average length of the longest increasing subsequences in random involutions avoiding 231 and another pattern which is a layered permutation. We obtain an explicit formula of the generating function and apply it to exemplify a set of interesting examples, which extend recent results of the first author with Yıldırım [Turkish J. Math. 43 (2019) 2183–2192], where the longest increasing subsequences in involutions avoiding a pai...
The main theme of this paper is the enumeration of the occurrence of a pattern in words and permu... more The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence f v r (k, n), the number of n-array k-ary words that contain a given pattern v exactly r times. In addition, we study the asymptotic behavior of the random variable X n , the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity P (X n = r) = 1 k n f v r (k, n). In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence (f v r (k, n)) 1/n converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of X n , including a CLT, large deviation estimates, and the exact growth rate of the entropy of X n. Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.
We study the longest increasing subsequence problem for random permutations avoiding the pattern ... more We study the longest increasing subsequence problem for random permutations avoiding the pattern 312 and another pattern τ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern τ is monotone increasing or decreasing, or any pattern of length four.
A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)... more A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots(k-1)(k+1),$$ $$k(k+2)(k+3)\cdots(2k-1)(2k)12\cdots(k-1)(k+1),$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots k,$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)(2k)123\cdots k.$$ The $321$-$4$-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincaré polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases $k=2,3,4$. In this paper, we extend these results by finding an explicit expression for the generating function for the number of $321$-$k$-gon-avoiding permutations on $n$ letters. The generating function is expressed via Chebyshev polynomials of the second kind.
We find generating functions for the number of words avoiding certain patterns or sets of pattern... more We find generating functions for the number of words avoiding certain patterns or sets of patterns with at most 2 distinct letters and determine which of them are equally avoided. We also find exact numbers of words avoiding certain patterns and provide bijective proofs for the resulting formulae.
We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers.... more We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.
Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths ... more Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restricted permutations in which the pattern is any one of the six patterns of length three. Our methodology uses recurrences, generating functions, and complex analysis, and also yields more detailed information. Several ideas for future research are listed.
In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say tha... more In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.
We study the longest increasing subsequences in random involutions that avoid the patterns of len... more We study the longest increasing subsequences in random involutions that avoid the patterns of length three under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes.
We consider a stack sorting algorithm where only the appropriate output values are popped from th... more We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the 2-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of (t+1)reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length n was discovered along with a bijection between this family and the collection of alternating permutations of length n − 1. We calculate generating functions for the number permutations of length n and exact rev-tier t.
Discrete Mathematics, Algorithms and Applications, 2018
We consider the number of passes a permutation needs to take through a stack if we only pop the a... more We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation [Formula: see text] to be [Formula: see text]-pass sortable if [Formula: see text] is sortable using [Formula: see text] passes through the stack. Permutations that are [Formula: see text]-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of [Formula: see text]-pass sortable permutations in terms of their basis. We also show all [Formula: see text]-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer [Formula: see text]. Finally, we define the notion of tier of a permutation [Formula: see text] to be the minimum number of passes after the first pass required to sort [Formula: see text]. We then give a bijection between the class ...
Journal of Difference Equations and Applications, 2014
We prove that the generating function for the number of flattened permutations having a given num... more We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method. Contents 1. Introduction 1 2. Counting 13-2 patterns in flattened permutations 2 3. Flattened permutations with r occurrences of the pattern 13-2 6 References 15 Appendix A. Proof of Corollary 2.4 16 Appendix B. Proof of Lemma 3.5 16 Appendix C. The bounds for deg x c r,i (x) are sharp 18
We introduce a new equivalence relation on permutations where two permutations are equivalent if ... more We introduce a new equivalence relation on permutations where two permutations are equivalent if and only if they coincide on their excedance sets. This paper studies equivalence classes for several subsets of permutations. Enumerating results are presented for permutations, cycles and permutations avoiding one or two patterns of length three. Also, an open question is proposed.
We count the number of histories between the two degenerate minimum energy configurations of the ... more We count the number of histories between the two degenerate minimum energy configurations of the Ising model on a chain, as a function of the length n and the number d of kinks that appear above the critical temperature. This is equivalent to count permutations of length n avoiding certain subsequences depending on d. We give explicit generating functions and compute the asymptotics. The setting considered has a role when describing dynamics induced by quantum Hamiltonians with deconfined quasi-particles.
We count the number of occurrences of certain patterns in given words. We choose these words to b... more We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a morphism generating a nonrepetitive sequence.
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first ... more In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the n-th large Schröder number and thus one obtains a new bivariate refinement of these numbers as a consequence. We make use of auxiliary combinatorial statistics, special generating functions (specific to each class) and the kernel method to establish our results. In some cases, we utilize the conjecture itself in a creative way to aid in solving the system of functional equations satisfied by the associated generating functions.
In this paper, we compute the distribution of the first letter statistic on nine avoidance classe... more In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schröder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.
ABSTRACTWe introduce Peano words, which are words corresponding to finite approximationsof the Pea... more ABSTRACTWe introduce Peano words, which are words corresponding to finite approximationsof the Peano space filling curve. We then find the number of occurrences of certainpatterns in these words.Keywords: Combinatorics on words, the Peano curve, generalized pattern 1. Introduction and BackgroundWe write permutations as words π = a 1 a 2 ···a n , whose letters are distinct and usuallyconsist of the integers 1,2,...,n.An occurrence of a pattern p in a permutation π is “classically” defined as a sub-sequence in π (of the same length as the length of p) whose letters are in the samerelative order as those in p. Formally speaking, for r ≤ n, we say that a permutationσ in the symmetric group S n has an occurrence of the pattern p ∈ S r if there exist1 ≤ i 1 < i 2 < ··· < i r ≤ n such that p = σ(i 1 )σ(i 2 )...σ(i r ) in reduced form. Thereduced form of a permutation σ on a set {j 1 ,j 2 ,...,j r }, where j 1 < j 2 < ··· < j r , isa permutation σ 1 obtained by renaming the le...
The problems of deciding the length of the longest increasing subsequence for a permutation or an... more The problems of deciding the length of the longest increasing subsequence for a permutation or an involution, the expectation and the limiting distribution, surveyed in Stanley’s ICM (International Congress of Mathematicians) Plenary address, have been widely studied by combinatorialists, analysts and probabilists. Partially motivated by the intriguing phenomenon stated by Simion and Schmidt [European J. Combin. 6 (1985) 383–406] that 231-avoiding permutations are exactly the set of layered permutations, in this paper, we investigate the limiting behavior of the average length of the longest increasing subsequences in random involutions avoiding 231 and another pattern which is a layered permutation. We obtain an explicit formula of the generating function and apply it to exemplify a set of interesting examples, which extend recent results of the first author with Yıldırım [Turkish J. Math. 43 (2019) 2183–2192], where the longest increasing subsequences in involutions avoiding a pai...
The main theme of this paper is the enumeration of the occurrence of a pattern in words and permu... more The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence f v r (k, n), the number of n-array k-ary words that contain a given pattern v exactly r times. In addition, we study the asymptotic behavior of the random variable X n , the number of pattern occurrences in a random n-array word. The two topics are closely related through the identity P (X n = r) = 1 k n f v r (k, n). In particular, we show that for any r ≥ 0, the Stanley-Wilf sequence (f v r (k, n)) 1/n converges to a limit independent of r, and determine the value of the limit. We then obtain several limit theorems for the distribution of X n , including a CLT, large deviation estimates, and the exact growth rate of the entropy of X n. Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations.
We study the longest increasing subsequence problem for random permutations avoiding the pattern ... more We study the longest increasing subsequence problem for random permutations avoiding the pattern 312 and another pattern τ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern τ is monotone increasing or decreasing, or any pattern of length four.
A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)... more A $321$-$k$-gon-avoiding permutation $\pi$ avoids $321$ and the following four patterns: $$k(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots(k-1)(k+1),$$ $$k(k+2)(k+3)\cdots(2k-1)(2k)12\cdots(k-1)(k+1),$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)1(2k)23\cdots k,$$ $$(k+1)(k+2)(k+3)\cdots(2k-1)(2k)123\cdots k.$$ The $321$-$4$-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincaré polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases $k=2,3,4$. In this paper, we extend these results by finding an explicit expression for the generating function for the number of $321$-$k$-gon-avoiding permutations on $n$ letters. The generating function is expressed via Chebyshev polynomials of the second kind.
We find generating functions for the number of words avoiding certain patterns or sets of pattern... more We find generating functions for the number of words avoiding certain patterns or sets of patterns with at most 2 distinct letters and determine which of them are equally avoided. We also find exact numbers of words avoiding certain patterns and provide bijective proofs for the resulting formulae.
We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers.... more We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of generating trees to show that the patterns 12132, 12123, 12321, 12231, 12213 are all equivalent to the pattern $12312$.
Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths ... more Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restricted permutations in which the pattern is any one of the six patterns of length three. Our methodology uses recurrences, generating functions, and complex analysis, and also yields more detailed information. Several ideas for future research are listed.
In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say tha... more In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.
We study the longest increasing subsequences in random involutions that avoid the patterns of len... more We study the longest increasing subsequences in random involutions that avoid the patterns of length three under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes.
We consider a stack sorting algorithm where only the appropriate output values are popped from th... more We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the 2-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of (t+1)reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length n was discovered along with a bijection between this family and the collection of alternating permutations of length n − 1. We calculate generating functions for the number permutations of length n and exact rev-tier t.
Discrete Mathematics, Algorithms and Applications, 2018
We consider the number of passes a permutation needs to take through a stack if we only pop the a... more We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation [Formula: see text] to be [Formula: see text]-pass sortable if [Formula: see text] is sortable using [Formula: see text] passes through the stack. Permutations that are [Formula: see text]-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of [Formula: see text]-pass sortable permutations in terms of their basis. We also show all [Formula: see text]-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer [Formula: see text]. Finally, we define the notion of tier of a permutation [Formula: see text] to be the minimum number of passes after the first pass required to sort [Formula: see text]. We then give a bijection between the class ...
Journal of Difference Equations and Applications, 2014
We prove that the generating function for the number of flattened permutations having a given num... more We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method. Contents 1. Introduction 1 2. Counting 13-2 patterns in flattened permutations 2 3. Flattened permutations with r occurrences of the pattern 13-2 6 References 15 Appendix A. Proof of Corollary 2.4 16 Appendix B. Proof of Lemma 3.5 16 Appendix C. The bounds for deg x c r,i (x) are sharp 18
We introduce a new equivalence relation on permutations where two permutations are equivalent if ... more We introduce a new equivalence relation on permutations where two permutations are equivalent if and only if they coincide on their excedance sets. This paper studies equivalence classes for several subsets of permutations. Enumerating results are presented for permutations, cycles and permutations avoiding one or two patterns of length three. Also, an open question is proposed.
We count the number of histories between the two degenerate minimum energy configurations of the ... more We count the number of histories between the two degenerate minimum energy configurations of the Ising model on a chain, as a function of the length n and the number d of kinks that appear above the critical temperature. This is equivalent to count permutations of length n avoiding certain subsequences depending on d. We give explicit generating functions and compute the asymptotics. The setting considered has a role when describing dynamics induced by quantum Hamiltonians with deconfined quasi-particles.
We count the number of occurrences of certain patterns in given words. We choose these words to b... more We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a morphism generating a nonrepetitive sequence.
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