Pattern avoidance in non-crossing and non-nesting permutations
Kassie Archer, Robert P. Laudone
TL;DR
This work studies pattern avoidance in two Stirling-variation permutation families—non-crossing and non-nesting—focusing on single 3-pattern avoidance (notably $231$ and its symmetries) and, for non-crossing, $122$-avoidance. The authors derive implicit generating-function equations for the counting sequences, obtaining $P(x)$ satisfying $x^3P(x)^3-(x^3+3x^2+x)P(x)^2+(2x^2-x+1)P(x)+x-1=0$ for non-nesting and $\bar P(x)$ satisfying $x^2\bar P(x)^4-(x^2+x)\bar P(x)^3-x\bar P(x)^2+(x+1)\bar P(x)-1=0$ for non-crossing, along with explicit series expansions and growth-rate data. They also classify $122$-avoidance in the non-crossing case, giving exact counts for joint avoidance with patterns in $\mathcal S_3$ (e.g., $\bar q_n(122)=C_n$, $\bar q_n(122,213)=F_n$, $\bar q_n(122,231)=2^{n-1}$, etc.). The results reveal distinct algebraic generating functions and growth behaviors tied to labeled-matchings and Catalan-structure combinatorics, and they conclude with open questions on $321$-avoidance and broader pattern-avoidance in multisets and trees.
Abstract
Non-crossing and non-nesting permutations are variations of the well-known Stirling permutations. A permutation $π$ 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).