Related papers: A positional statistic for 1324-avoiding permutati…

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…

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. Furthermore, let $\mid A(n)\mid$ denote the cardinality of the set $A(n)=A\cap [n]$. The limit…

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$…

Let $S_{\rm lcm}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,\pi(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $\pi$ of $[n]$ such that…

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…

For each positive integer $k$, we consider five well-studied posets defined on the set of Dyck paths of semilength $k$. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets.…

In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;\tau)$ of all permutations in $S_n$ that contain $\tau\in S_k$ exactly once and avoid a subset…

We study the number of 231-avoiding permutations with $j$-descents and maximum drop is less than or equal to $k$ which we denote by $a_{n,231,j}^{(k)}$. We show that $a_{n,231,j}^{(k)}$ also counts the number of Dyck paths of length $2n$…

Define a permutation $\sigma$ to be coprime if $\gcd(m,\sigma(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c =…

We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial…

A permutation array $A$ is a set of permutations on a finite set $\Omega$, say of size $n$. Given distinct permutations $\pi, \sigma\in \Omega$, we let $hd(\pi, \sigma) = |\{ x\in \Omega: \pi(x) \ne \sigma(x) \}|$, called the Hamming…

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…

Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum…

A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid $[n]^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to…

Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an…

This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation $\sg = \sg_1 ... \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the quadrant…

We give a positive answer to a question raised by Davis et al. ({\em Discrete Mathematics} 341, 2018), concerning permutations with the same pinnacle set. Given $\pi\in S_n$, a {\em pinnacle} of $\pi$ is an element $\pi_i$ ($i\neq 1,n$)…

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined…

For a fixed integer $m\ge 1$, let $\mathcal{A}_n^{(m)}$ be the set of permutations $\pi\in S_n$ that avoid the pattern $132$ and satisfy the adjacency bound $|\pi_{i+1}-\pi_i|\le m$ for all $i$. Here, a pattern $132$ means three indices…

Stankova and West showed that for any non-negative integer $s$ and any permutation $\gamma$ of $\{4,5,\dots,s+3\}$ there are as many permutations that avoid $231\gamma$ as there are that avoid $312\gamma$. We extend this result to the…