Related papers: On the poset of King-Non-Attacking permutations

A sequence $f\colon\{1,\dots,n\}\to\mathbb{R}$ contains a permutation $\pi$ of length $k$ if there exist $i_1<\dots<i_k$ such that, for all $x,y$, $f(i_x)<f(i_y)$ if and only if $\pi(x)<\pi(y)$; otherwise, $f$ is said to be $\pi$-free. In…

In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this…

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to…

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We…

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 $\pi_i$ ($i\neq 1,n$)…

The technique of in-situ associative permuting is introduced which is an association of in-situ permuting and in-situ inverting. It is suitable for associatively permutable permutations of {1,2,...,n} where the elements that will be…

A subset $S$ of the alternating group on $n$ points is {\it intersecting} if for any pair of permutations $\pi,\sigma$ in $S$, there is an element $i\in \{1,\dots,n\}$ such that $\pi(i)=\sigma(i)$. We prove that if $S$ is intersecting, then…

The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the…

An alternating permutation of length $n$ is a permutation $\pi=\pi_1 \pi_2 ... \pi_n$ such that $\pi_1 < \pi_2 > \pi_3 < \pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(\sigma)$ be set of…

A permutation $\pi$ of a multiset is said to be a {\em quasi-Stirling } permutation if there does not exist four indices $i<j<k<\ell$ such that $\pi_i=\pi_k$ and $\pi_j=\pi_{\ell}$. Define $$ \overline{Q}_{\mathcal{M}}(t,u,v)=\sum_{\pi\in…

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(\pi_1,\ldots,\pi_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive sums of $\pi$, i.e., $s_i:=\sum_{j=0}^{k-1}\pi_{i+j}$ for $i=1,\ldots,n$, where we let…

We say a permutation $\pi=\pi_1\pi_2\cdots\pi_n$ in the symmetric group $\mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1}<\pi_i>\pi_{i+1}$ and we let $P(\pi)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$ is a peak of $\pi$}\}$.…

In the first paper in this series we estimated the probability that a random permutation $\pi\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\pi$ has $m$…

A pure excedance in a permutation $\pi=\pi_1\pi_2\ldots \pi_n$ is a position $i<\pi_i$ such that there is no $j<i$ with $i\leq \pi_j<\pi_i$. We present a one-to-one correspondence on the symmetric group that transports pure excedances to…

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…

A Parity Alternating Permutation of the set $[n] = \{1, 2,\ldots, n\}$ is a permutation with even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the…

In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our…

For a permutation $\pi$ the major index of $\pi$ is the sum of all indices $i$ such that $\pi_i > \pi_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In…

There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and…

\noindent In our contribution to this volume we deal with \emph{discrete} symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In…