Pattern avoidance in non-crossing and non-nesting permutations

TL;DR

This paper studies pattern avoidance in special classes of permutations called non-crossing and non-nesting permutations, providing generating functions for those avoiding certain length-3 patterns, advancing enumeration methods in combinatorics.

Contribution

It introduces generating functions for non-crossing and non-nesting permutations avoiding specific length-3 patterns, solving an open enumeration problem.

Findings

Derived generating functions for pattern-avoiding permutations

Solved enumeration for patterns 231, 132, 213, 312

Extended understanding of pattern avoidance in Stirling permutations

Abstract

Non-crossing and non-nesting permutations are variations of the well-known Stirling permutations. A permutation $\pi$ on $\{1,1,2,2,\ldots, n,n\}$ is called non-crossing if it avoids the crossing patterns $\{1212,2121\}$ and is called non-nesting if it avoids the nesting patterns $\{1221,2112\}.$ Pattern avoidance in these permutations has been considered in recent years, but it has remained open to enumerate the non-crossing and non-nesting permutations that avoid a single pattern of length 3. In this paper, we provide generating functions for those non-crossing and non-nesting permutations that avoid the pattern 231 (and, by symmetry, the patterns 132, 213, or 312).