Descent generating polynomials for ($n-3$)- and ($n-4$)-stack-sortable (pattern-avoiding) permutations

TL;DR

This work determines the descent generating polynomials for $(n-3)$- and $(n-4)$-stack-sortable permutations and their pattern-avoiding variants, expressing them in terms of Eulerian polynomials $A_n(x)$ and Narayana polynomials $N_n(x)$. By leveraging West’s classifications and the Claesson–Dukes–Steingrímsson tables, the authors derive explicit closed forms for $W_{n,n-3}(x)$ and $W_{n,n-4}(x)$, and provide closed expressions for $W^{p}_{n,n-2}(x)$, $W^{p}_{n,n-3}(x)$, and $W^{p}_{n,n-4}(x)$ for all length-3 patterns $p$, with the latter written as combinations of $A_n(x)$ and $N_n(x)$. The results unify and extend prior enumeration results, and reveal relationships between descent distributions and classical polynomials, yielding new conjectures linking $213$- and $321$-avoiding cases across all stack-sorting levels. The formulas offer avenues to study analytic properties such as unimodality and real-rootedness and hint at connections to restricted-stack machines, deepening the combinatorial understanding of pattern-avoiding stack-sortable permutations.

Abstract

In this paper, we find distribution of descents over $(n-3)$- and $(n-4)$-stack-sortable permutations in terms of Eulerian polynomials. Our results generalize the enumeration results by Claesson, Dukes, and Steingrímsson on $(n-3)$- and $(n-4)$-stack-sortable permutations. Moreover, we find distribution of descents on $(n-2)$-, $(n-3)$- and $(n-4)$-stack-sortable permutations that avoid any given pattern of length 3, which extends known results in the literature on distribution of descents over pattern-avoiding 1- and 2-stack-sortable permutations. Our distribution results also give enumeration of $(n-2)$-, $(n-3)$- and $(n-4)$-stack-sortable permutations avoiding any pattern of length 3. One of our conjectures links our work to stack-sorting with restricted stacks, and the other conjecture states that 213-avoiding permutations sortable with $t$ stacks are equinumerous with 321-avoiding permutations sortable with $t$ stacks for any $t$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Remark 9
• Lemma 10