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

Combinatorial properties of permutation tableaux

We give another construction of a permutation tableau from its corresponding permutation and construct a permutation-preserving bijection between $1$-hinge and $0$-hinge tableaux. We also consider certain alignment and crossing statistics on permutation tableaux that have previously been shown to be equidistributed by mapping them to patterns in related permutations. We give two direct maps on tableaux that prove the equidistribution of those statistics by exchanging some statistics and preserving the rest. Finally, we enumerate some sets of permutations that are restricted both by pattern avoidance and by certain parameters of their associated permutation tableaux

Permutation patterns, Stanley symmetric functions, and the Edelman-Greene correspondence

Generalizing the notion of a vexillary permutation, we introduce a filtration of $S_{\infty}$ by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by avoiding a finite set of patterns. In doing so, we show that if $w$ is a permutation containing $v$ as a pattern, then there is an injection from the set of Edelman-Greene tableaux of $v$ to the set of Edelman-Greene tableaux of $w$ which respects inclusion of shapes. We also consider the set of permutations whose Edelman-Greene tableaux have distinct shapes, and show that it is closed under taking patterns

Asymptotics for minimal overlapping patterns for generalized Euler permutations, standard tableaux of rectangular shape, and column strict arrays

A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal overlapping patterns in $S_j$ is at least $3 -e$. Given a permutation $\sigma$, we let $\text{Des}(\sigma)$ denote the set of descents of $\sigma$. We study the class of permutations $\sigma \in S_{kn}$ whose descent set is contained in the set $\{k,2k, \ldots (n-1)k\}$. For example, up-down permutations in $S_{2n}$ are the set of permutations whose descent equal $\sigma$ such that $\text{Des}(\sigma) = \{2,4, \ldots, 2n-2\}$. There are natural analogues of the minimal overlapping permutations for such classes of permutations and we study the proportion of minimal overlapping patterns for each such class. We show that the proportion of minimal overlapping permutations in such classes approaches $1$ as $k$ goes to infinity. We also study the proportion of minimal overlapping patterns in standard Young tableaux of shape $(n^k)$.Comment: Accepted by Discrete Math and Theoretical Computer Science. Thank referees' for their suggestion

Bijections for Permutation Tableaux

In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6]

RSK tableaux and the weak order on fully commutative permutations

For each fully commutative permutation, we construct a "boolean core," which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order.Comment: 20 pages, 2 figure