Positional Marked Patterns in Permutations

TL;DR

This paper introduces positional marked patterns in permutations, analyzing their distribution and showing certain pattern pairs have identical distributions, revealing new combinatorial properties.

Contribution

It defines a new class of permutation statistics based on positional marked patterns and proves equidistribution results for specific pattern pairs.

Findings

Patterns 1 extunderscore 2 3 and 1 extunderscore 3 2 are equidistributed.

Equidistribution extends to various pattern collections and lengths.

New combinatorial properties of permutation patterns are established.

Abstract

We define and study positional marked patterns, permutations $\tau$ where one of elements in $\tau$ is underlined. Given a permutation $\sigma$, we say that $\sigma$ has a $\tau$-match at position $i$ if $\tau$ occurs in $\sigma$ in such a way that $\sigma_i$ plays the role of the underlined element in the occurrence. We let $pmp_\tau(\sigma)$ denote the number of positions $i$ which $\sigma$ has a $\tau$-match. This defines a new class of statistics on permutations, where we study such statistics and prove a number of results. In particular, we prove that two positional marked patterns $1\underline{2}3$ and $1\underline{3}2$ give rise to two statistics that have the same distribution. The equidistibution phenomenon also occurs in other several collections of patterns like $\left \{1\underline{2}3 , 1\underline{3}2 \right \}$, and $\left \{ 1\underline234, 1\underline243, \underline2134, \underline2 1 4 3 \right \}$, as well as two positional marked patterns of any length $n$: $\left \{ 1\underline 2\tau , \underline 21\tau \right \}$.