Crossings over permutations avoiding some pairs of patterns of length three

TL;DR

This paper analyzes the distribution of crossings in permutations avoiding specific pairs of length-three patterns, providing new combinatorial interpretations and relationships between different pattern-avoiding classes.

Contribution

It introduces new combinatorial interpretations of known triangles via crossings in pattern-avoiding permutations and explores relationships between distributions for different pattern pairs.

Findings

Distribution formulas for crossings avoiding {321,231}, {123,132}, {123,213}

New combinatorial interpretations of known triangles

Relationships between distributions for pattern pairs involving 231 and 312

Abstract

In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs $\{321,231\}$, $\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorial interpretations of two known triangles in terms of restricted permutations statistic. For other pairs of patterns of length three, we find relationships between the polynomial distributions of the crossings over permutations that avoid the pairs containing the pattern 231 on the first hand and the pattern 312 on the other hand.