Sittipong Thamrongpairoj ; Jeffrey B. Remmel - Positional Marked Patterns in Permutations

dmtcs:7171 - Discrete Mathematics & Theoretical Computer Science, August 8, 2022, vol. 24, no. 1 -
Positional Marked Patterns in Permutations

Authors: Sittipong Thamrongpairoj ; Jeffrey B. Remmel

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 $n$: $\left \{ 1\underline 2\tau , \underline 21\tau \right \}$.

Volume: vol. 24, no. 1
Section: Combinatorics
Published on: August 8, 2022
Accepted on: June 2, 2022
Submitted on: February 12, 2021
Keywords: Mathematics - Combinatorics,05A05


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