We present enumerations of inversion sequences that avoid 120, 210, and a pattern of length three... more We present enumerations of inversion sequences that avoid 120, 210, and a pattern of length three. More precisely, we characterize the rules of generating trees and we write the corresponding function equations of these inversion sequences.
Proceedings of the American Mathematical Society, Jul 9, 2008
We complete the Wilf classification of signed patterns of length 5 for both signed permutations a... more We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S 5 , S 6 , and S 7 for involutions.
In this paper we study the generating polynomials obtained by enumerating signed simsun permutati... more In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
The art of discrete and applied mathematics, Jul 31, 2023
Let S n be the set of all permutations of {1, 2, . . . , n} represented in cycle notation. Define... more Let S n be the set of all permutations of {1, 2, . . . , n} represented in cycle notation. Define a n,m to be the number of π ∈ S n such that the length of the initial longest increasing sequence (ILIS) in π is at most m. For fixed m, we find the exponential generating function for the sequence a n,m , and give an asymptotic formula for a n,m when n → ∞. Moreover, we show that the expected value of the total ILIS in all cycles of π over all permutations of S n is asymptotic to e n j=1 1 j .
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 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.
The purpose of this paper is to investigate the connection between the Lotka-Volterra system and ... more The purpose of this paper is to investigate the connection between the Lotka-Volterra system and combinatorics. We study several context-free grammars associated with the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers of the second kind and Eulerian numbers, are generated by these context-free grammars. In particular, we present grammatical characterization of some statistics on cyclically ordered partitions.
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutati... more Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 25, 2022
We study the enumeration of inversion sequences that avoid pattern 021 and another pattern of len... more We study the enumeration of inversion sequences that avoid pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of d-regular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.
The art of discrete and applied mathematics, Mar 7, 2023
A permutation pattern in which all letters within an occurrence are required to be adjacent is kn... more A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing partitions of size n, denoted by N C n , and derive formulas for the generating functions of these distributions on N C n . As special cases of our results, we obtain formulas for the generating functions enumerating the members of N C n according to the number of occurrences of any subword of length three with distinct letters. Simple expressions for the total number of occurrences of a pattern over all members of N C n are also deduced. Some connections are made with the related problem of counting Dyck paths according to the number of occurrences of certain types of strings. Further, for the subwords 12 • • • m and 213 • • • m where m ≥ 3, we consider the joint distribution with the descents statistic and make use of the kernel method to establish the results in these cases.
In this paper, we study the generating functions for the number of pattern restricted Stirling pe... more In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.
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.
We revisit staircases for words and prove several exact as well as asymptotic results for longest... more We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence h r,k (n), the number of n-array words with r separations over alphabet [k] and show that for any r ≥ 0, the growth sequence (h r,k (n)) 1/n converges to a characterized limit, independent of r. In addition, we study the asymptotic behavior of the random variable S k (n), the number of staircase separations in a random word in [k] n and obtain several limit theorems for the distribution of S k (n), including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of S k (n). Finally, we obtain similar results, including growth limits, for longest L-staircase subwords and subsequences.
A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: The 321-4-gon-avoi... more A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: 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.
Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric... more Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [BS]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no hyphens (that allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p.
HAL (Le Centre pour la Communication Scientifique Directe), 2008
Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing pari... more Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N, (N1, . . . , Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = {1, . . . , k}. In particular, we refine and generalize some of the results by Burstein and Mansour.
In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on... more In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on n letters. This conjecture was proved analytically by Zeilberger in [Z]. Later, Dulucq, Gire, and Guibert [DGG] gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ on k letters. In several interesting cases this generating function can be expressed in terms of the generating function for the Fibonacci numbers or the generating function for the Pell numbers.
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutati... more Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.
We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 no... more We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. The insertion encoding algorithm (INSENC) accounts for many of these and some others have been previously counted; in this paper, we find the generating function for each of the remaining 36 triples and it turns out to be algebraic in every case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A particularly nice so-called cell decomposition is used in one case and a bijection is used for another.
)|τ ∈ S n , 1 ≤ a i ≤ r} be the set of all signed permutations on the symbols 1, 2, . . . , n wit... more )|τ ∈ S n , 1 ≤ a i ≤ r} be the set of all signed permutations on the symbols 1, 2, . . . , n with signs 1, 2, . . . , r. We prove, for every 2-letter signed pattern [τ ] a , that the number of [τ ] a -avoiding signed permutations in E r n is given by the formula n j=0 j!(r -1) j n j 2 . Also we prove that there are only one Wilf class for r = 1, four Wilf classes for r = 2, and six Wilf classes for r ≥ 3.
The past decade has seen a flurry of research into pattern avoiding permutations but little of it... more The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.
We present enumerations of inversion sequences that avoid 120, 210, and a pattern of length three... more We present enumerations of inversion sequences that avoid 120, 210, and a pattern of length three. More precisely, we characterize the rules of generating trees and we write the corresponding function equations of these inversion sequences.
Proceedings of the American Mathematical Society, Jul 9, 2008
We complete the Wilf classification of signed patterns of length 5 for both signed permutations a... more We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S 5 , S 6 , and S 7 for involutions.
In this paper we study the generating polynomials obtained by enumerating signed simsun permutati... more In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
The art of discrete and applied mathematics, Jul 31, 2023
Let S n be the set of all permutations of {1, 2, . . . , n} represented in cycle notation. Define... more Let S n be the set of all permutations of {1, 2, . . . , n} represented in cycle notation. Define a n,m to be the number of π ∈ S n such that the length of the initial longest increasing sequence (ILIS) in π is at most m. For fixed m, we find the exponential generating function for the sequence a n,m , and give an asymptotic formula for a n,m when n → ∞. Moreover, we show that the expected value of the total ILIS in all cycles of π over all permutations of S n is asymptotic to e n j=1 1 j .
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 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.
The purpose of this paper is to investigate the connection between the Lotka-Volterra system and ... more The purpose of this paper is to investigate the connection between the Lotka-Volterra system and combinatorics. We study several context-free grammars associated with the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers of the second kind and Eulerian numbers, are generated by these context-free grammars. In particular, we present grammatical characterization of some statistics on cyclically ordered partitions.
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutati... more Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 25, 2022
We study the enumeration of inversion sequences that avoid pattern 021 and another pattern of len... more We study the enumeration of inversion sequences that avoid pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of d-regular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.
The art of discrete and applied mathematics, Mar 7, 2023
A permutation pattern in which all letters within an occurrence are required to be adjacent is kn... more A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing partitions of size n, denoted by N C n , and derive formulas for the generating functions of these distributions on N C n . As special cases of our results, we obtain formulas for the generating functions enumerating the members of N C n according to the number of occurrences of any subword of length three with distinct letters. Simple expressions for the total number of occurrences of a pattern over all members of N C n are also deduced. Some connections are made with the related problem of counting Dyck paths according to the number of occurrences of certain types of strings. Further, for the subwords 12 • • • m and 213 • • • m where m ≥ 3, we consider the joint distribution with the descents statistic and make use of the kernel method to establish the results in these cases.
In this paper, we study the generating functions for the number of pattern restricted Stirling pe... more In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.
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.
We revisit staircases for words and prove several exact as well as asymptotic results for longest... more We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number, the latter being defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence h r,k (n), the number of n-array words with r separations over alphabet [k] and show that for any r ≥ 0, the growth sequence (h r,k (n)) 1/n converges to a characterized limit, independent of r. In addition, we study the asymptotic behavior of the random variable S k (n), the number of staircase separations in a random word in [k] n and obtain several limit theorems for the distribution of S k (n), including a law of large numbers, a central limit theorem, and the exact growth rate of the entropy of S k (n). Finally, we obtain similar results, including growth limits, for longest L-staircase subwords and subsequences.
A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: The 321-4-gon-avoi... more A 321-k-gon-avoiding permutation π avoids 321 and the following four patterns: 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.
Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric... more Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [BS]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no hyphens (that allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p.
HAL (Le Centre pour la Communication Scientifique Directe), 2008
Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing pari... more Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N, (N1, . . . , Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = {1, . . . , k}. In particular, we refine and generalize some of the results by Burstein and Mansour.
In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on... more In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on n letters. This conjecture was proved analytically by Zeilberger in [Z]. Later, Dulucq, Gire, and Guibert [DGG] gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ on k letters. In several interesting cases this generating function can be expressed in terms of the generating function for the Fibonacci numbers or the generating function for the Pell numbers.
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutati... more Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.
We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 no... more We determine all 242 Wilf classes of triples of 4-letter patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (agree in the first 16 terms) are in fact identical. The insertion encoding algorithm (INSENC) accounts for many of these and some others have been previously counted; in this paper, we find the generating function for each of the remaining 36 triples and it turns out to be algebraic in every case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A particularly nice so-called cell decomposition is used in one case and a bijection is used for another.
)|τ ∈ S n , 1 ≤ a i ≤ r} be the set of all signed permutations on the symbols 1, 2, . . . , n wit... more )|τ ∈ S n , 1 ≤ a i ≤ r} be the set of all signed permutations on the symbols 1, 2, . . . , n with signs 1, 2, . . . , r. We prove, for every 2-letter signed pattern [τ ] a , that the number of [τ ] a -avoiding signed permutations in E r n is given by the formula n j=0 j!(r -1) j n j 2 . Also we prove that there are only one Wilf class for r = 1, four Wilf classes for r = 2, and six Wilf classes for r ≥ 3.
The past decade has seen a flurry of research into pattern avoiding permutations but little of it... more The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.
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Papers by Toufik Mansour