Positional Marked Patterns in Permutations

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 $pm{p}_{\tau }(\sigma )$ 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 $\left\{1\underline{2}34,1\underline{2}43,\underline{2}134,\underline{2}143\right\}$, as well as two positional marked patterns of any length $n$: $\left\{1\underline{2}\tau ,\underline{2}1\tau \right\}$.

References in corpus (1)

• Quadrant marked mesh patterns in alternating permutations