Stack-sorting preimages and 0-1-trees

TL;DR

This work connects stack-sorting preimages of pattern-avoiding permutation classes to a Boolean-Catalan combinatorial family via $(0,1)$-trees. By analyzing permutations through a decomposition $p=LnR$ and using two constructive assembly modes, the authors show that the counts for $|s^{-1}(\operatorname{Av}_n(132,312))|$ and $|s^{-1}(\operatorname{Av}_n(231,312))|$ satisfy the same recurrence $a_n=2a_{n-1}+2\sum_{i=2}^{n-1}a_{i-1}a_{n-i}$ with $a_1=1$, and share the generating function $A(z)=\frac{1-2z-\sqrt{1-4z-4z^2}}{4z}$. These results illuminate a unified combinatorial mechanism behind seemingly disparate preimage counts and motivate further exploration of the remaining third equality $a_n=|s^{-1}(\operatorname{Av}_n(132,231))|$ via kernel methods. The work highlights a robust link between stack-sorting, pattern-avoiding permutations, and Catalan-like structures.

Abstract

We define a class of partially labeled trees and use them to find simple proofs for two recent enumeration results of Colin Defant concerning stack-sorting preimages of permutation classes.

Figures (5)

• Figure 1: The six 0-1 trees on three vertices.
• Figure 2: A generic permutation avoiding 132 and 312.
• Figure 3: The two possible values of the leftmost entry of $s(R)$ in $p$.
• Figure 4: A generic permutation avoiding 231 and 312, called a layered permutation.
• Figure 5: The two ways of combining $s(L)$ and $s(R)$.

Theorems & Definitions (9)

• Definition 2.1
• Proposition 2.2
• proof
• Theorem 3.1
• proof
• Example 3.2
• Theorem 4.1
• proof
• Example 4.2