On multi-avoidance of generalized patterns

In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such $n$-permutations are $2^{n-1}$, the number of involutions in $\mathcal{S}_n$, and $2E_n$, where $E_n$ is the $n$-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form $x-y-z$ (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.Comment: 26 page

Generalized permutation patterns - a short survey

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns

Simultaneous avoidance of generalized patterns

In [BabStein] Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case. Also we consider permutations that avoid a pattern of the form $x-yz$ or $xy-z$ and begin with one of the patterns $12... k$, $k(k-1)... 1$, $23... k1$, $(k-1)(k-2)... 1k$ or end with one of the patterns $12... k$, $k(k-1)... 1$, $1k(k-1)... 2$, $k12... (k-1)$. For each of these cases we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides we generalize some of the obtained results as well as some of the results given in [Kit3]: we consider permutations avoiding certain generalized 3-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.Comment: 18 page

Introduction to Partially Ordered Patterns

We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution. Moreover, we discuss q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea

Inversion Polynomials for Permutations Avoiding Consecutive Patterns

In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $\Pi$ and $\Pi'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi$ is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi'$. In 2013, Cameron and Killpatrick gave the inversion generating function for Fibonacci tableaux which are in one-to-one correspondence with the set of permutations that simultaneously avoid the consecutive patterns $321$ and $312.$ In this paper, we use the language of Fibonacci tableaux to study the inversion generating functions for permutations that avoid $\Pi$ where $\Pi$ is a set of five or fewer consecutive permutation patterns. In addition, we introduce the more general notion of a strip tableaux which are a useful combinatorial object for studying consecutive pattern avoidance. We go on to give the inversion generating functions for all but one of the cases where $\Pi$ is a subset of three consecutive permutation patterns and we give several results for $\Pi$ a subset of two consecutive permutation patterns

Place-difference-value patterns: A generalization of generalized permutation and word patterns

Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson introduced the notion of a "generalized permutation pattern" (GP) which generalizes the concept of "classical" permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrimsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Melou et al. introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets. In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we raise several bijective questions linking our patterns to other combinatorial objects.Comment: 18 pages, 2 figures, 1 tabl

Generalized pattern avoidance with additional restrictions

Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a generalized 3-pattern with no dashes and whose k leftmost or k rightmost letters form either an increasing or decreasing subword. Moreover, we find a bijection between n-permutations that avoid the pattern 132 and begin with the pattern 12 and increasing rooted trimmed trees with n+1 nodes.Comment: 18 page

Asymptotic enumeration of permutations avoiding generalized patterns

Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns.Comment: 14 pages, 3 figures, to be published in Adv. in Appl. Mat