Related papers: Constraining Strong c-Wilf Equivalence Using Clust…

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…

Let S_n denote the symmetric group of all permutations of the set {1, 2, ...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of Pi in the…

A frequent topic in the study of pattern avoidance is identifying when two sets of patterns $\Pi, \Pi'$ are Wilf equivalent, that is, when $|\text{Av}_n(\Pi)| = |\text{Av}_n(\Pi')|$ for all $n$. In recent work of Dokos et al. the notion of…

This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter to simple permutations and prove that if $\sigma, \pi$ are…

For $\eta\in S_3$, let $S_n^{\text{av}(\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\eta$, and let $E_n^{\text{av}(\eta)}$ denote the expectation with respect to the uniform probability measure on…

We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345,12354} and {45123,45213} that extends the list of 10 related conjectures proved in the literature in a series of papers. To achieve our goals, we prove…

We give a sufficient condition for the two dashed patterns $\tau^{(1)}-\tau^{(2)}-\cdots-\tau^{(\ell)}$ and $\tau^{(\ell)}-\tau^{(\ell-1)}-\cdots-\tau^{(1)}$ to be (strongly) Wilf-equivalent. This permits to solve in a unified way several…

Savage and Sagan have recently defined a notion of st-Wilf equivalence for any permutation statistic st and any two sets of permutations $\Pi$ and $\Pi'$. In this paper we give a thorough investigation of st-Wilf equivalence for the charge…

Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $\pi$ and $\tau$, represented as sequences of integers, and the task is to determine whether $\tau$ contains a subsequence order-isomorphic to…

We extend the notion of shape-Wilf-equivalence to vincular patterns (also known as "generalized patterns" or "dashed patterns"). First we introduce a stronger equivalence on patterns which we call filling-shape-Wilf-equivalence. When…

Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…

For about 10 years, the classification of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n-1,n-2,n,tau)~(n-2,n,n-1,tau) for any tau in…

Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on…

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…

In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To…

A descent $k$ of a permutation $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$ is called a big descent if $\pi_{k}>\pi_{k+1}+1$; denote the number of big descents of $\pi$ by $\operatorname{bdes}(\pi)$. We study the distribution of the…

Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets…

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{\pi, \sigma}^{\lambda} ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(\alpha )}_\pi…

A permutation statistic $\operatorname{st}$ is said to be shuffle-compatible if the distribution of $\operatorname{st}$ over the set of shuffles of two disjoint permutations $\pi$ and $\sigma$ depends only on $\operatorname{st}\pi$,…

We consider the well-studied pattern counting problem: given a permutation $\pi \in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $\tau \in \mathbb{S}_k$ in $\pi$. Our first result…