Related papers: Periodic Points of Consecutive-Pattern-Avoiding St…

A matching of the set $[2n]=\{ 1,2,\ldots ,2n\}$ is a partition of $[2n]$ into blocks with two elements, i.e. a graph on $[2n]$ such that every vertex has degree one. Given two matchings $\sigma$ and $\tau$ , we say that $\sigma$ is a…

We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the…

Pop-Stack Sorting is an algorithm that takes a permutation as an input and sorts its elements. It consists of several steps. At one step, the algorithm reads the permutation it has to process from left to right and reverses each of its…

Chen and collaborators give a recursively defined bijection from 021-avoiding ascent sequences to 021-avoiding (aka 132-avoiding) permutations. Here we give an algorithmic bijection from 021-avoiding ascent sequences to Dyck paths. Our…

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new…

The 321,hexagon-avoiding (321-hex) permutations were introduced and studied by Billey and Warrington in as a class of elements of S_n whose Kazhdan-Lusztig and Poincare polynomials and the singular loci of whose Schubert varieties have…

We introduce a new method for encoding permutations as weighted independent sets in a family of graphs we call cores. The encoding allows us to enumerate (1324, 2143)-, (1234, 1324, 2143)-, (1234, 1324, 1432, 3214)-avoiding permutations…

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1, \dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and…

A \emph{set partition} of the set $[n]=\{1,...c,n\}$ is a collection of disjoint blocks $B_1,B_2,...c, B_d$ whose union is $[n]$. We choose the ordering of the blocks so that they satisfy $\min B_1<\min B_2<...b<\min B_d$. We represent such…

Problem 8.1 in Astaiza et. al. asks about the relationship between the cycle decomposition of a permutation $\sigma$ and that of its symmetric tensor power $\sigma ^{\odot k}$. In this paper, we investigate this question and give formulas…

We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map $s$. As a first application, we give a new proof of Zeilberger's formula for the number of 2-stack-sortable…

Theorems relating permutations with objects in other fields of mathematics are often stated in terms of avoided patterns. Examples include various classes of Schubert varieties from algebraic geometry (Billey and Abe 2013), commuting…

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving…

It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$…

We study a system of intervals $I_1,\ldots,I_k$ on the real line and a continuous map $f$ with $f(I_1 \cup I_2 \cup \ldots \cup I_k)\supseteq I_1 \cup I_2 \cup \ldots \cup I_k$. It's conjectured that there exists a periodic point of period…

In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined…

Let T_k^m={\sigma \in S_k | \sigma_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any \tau in T_k^1 (or any \tau in T_k^k), the number of…

Egge and Mansour have recently studied permutations which avoid 1243 and 2143 regarding the occurrence of certain additional patterns. Some of the open questions related to their work can easily be answered by using permutation diagrams.…