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Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the…
We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of…
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice…
Let $\mathcal{S}_n(\pi)$ (resp. $\mathcal{I}_n(\pi)$ and $\mathcal{AI}_n(\pi)$) denote the set of permutations (resp. involutions and alternating involutions) of length $n$ which avoid the permutation pattern $\pi$. For $k,m\geq 1$,…
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…
Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the…
Vincular or dashed patterns resemble classical patterns except that some of the letters within an occurrence are required to be adjacent. We prove several infinite families of Wilf-equivalences for k-ary words involving vincular patterns…
Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying…
In this paper, we find explicit formulas or generating functions for the cardinalities of the sets $S_n(T,\tau)$ of all permutations in $S_n$ that avoid a pattern $\tau\in S_k$ and a set $T$, $|T|\geq 2$, of patterns from $S_3$. The main…
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0):…
Let $\sigma$ and $\tau$ be patterns of length three; that is $\sigma, \tau \in \{123,132,213,231,312,321\}$. In this paper, we enumerate the set of cyclic permutations in $\mathcal{S}_n$ that avoid $\sigma$ in their one-line notation and…
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…
Inversion sequences are finite sequences of non-negative integers, where the value of each entry is bounded from above by its position. Patterns in inversion sequences have been studied by Corteel-Martinez-Savage-Weselcouch and…
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…
A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of…
In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…
The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of…
Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when…