[Paper Review] Stack-Sorting Preimages of Permutation Classes

Colin Defant|arXiv (Cornell University)|Sep 10, 2018

Advanced Combinatorial Mathematics24 references12 citations

TL;DR

This paper introduces a novel framework for computing the preimages of permutation classes under West's stack-sorting map using valid hook configurations, enabling exact enumeration of permutations mapping to pattern-avoiding sets. It resolves open problems on 2-stack-sortable permutations and provides closed forms and generating functions for preimages of various classes, including unimodal sequences and symmetric patterns.

ABSTRACT

We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$. First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(π)|$ for any permutation $π$, answering a question of Bousquet-Mélou. We then enumerate permutations in various sets of the form $s^{-1}( ext{Av}(τ^{(1)},\ldots,τ^{(r)}))$, where $ ext{Av}(τ^{(1)},\ldots,τ^{(r)})$ is the set of permutations avoiding the patterns $τ^{(1)},\ldots,τ^{(r)}$. These preimage sets often turn out to be permutation classes themselves, so the current paper represents a new approach, based on the theory of valid hook configurations, for solving classical enumerative problems. In one case, we solve a problem previously posed by Bruner. We are often able to refine our counts by enumerating these permutations according to their number of descents or peaks. Our investigation not only provides several new combinatorial interpretations and identities involving known sequences, but also paves the way for several new enumerative problems.

Motivation & Objective

• To develop a general method for computing the size of the preimage $|s^{-1}( ext{Av}( au^{(1)}, reak au^{(2)}, reak \dots, reak au^{(r)}))|$ under West's stack-sorting map.
• To extend classical enumerative results on stack-sorting by characterizing preimages of permutation classes beyond sortable and 2-stack-sortable permutations.
• To refine enumeration by descent and peak statistics using valid hook configurations, providing deeper combinatorial structure.
• To resolve open problems on 2-stack-sortable permutations and propose new conjectures on preimage generating functions.
• To establish connections between preimage sets and known combinatorial sequences, including Catalan and Motzkin-like numbers.

Proposed method

• Introduces a theorem enabling efficient computation of $|s^{-1}( au)|$ for any permutation $\tau$, resolving a question posed by Bousquet-Mélou.
• Applies the theory of valid hook configurations to count permutations in $s^{-1}( ext{Av}( au^{(1)}, \dots, \tau^{(r)}))$ with respect to descent and peak statistics.
• Uses recursive structure of the stack-sorting map $s(\pi) = s(L)s(R)n$ to decompose preimage sets based on the position of the maximum element.
• Employs generating functions and combinatorial species techniques to derive closed forms for preimage counts, especially for $\text{Av}(132, 312)$, $\text{Av}(132, 231)$, and $\text{Av}(231, 321)$.
• Leverages known bijections (e.g., between fighting fish and 2-stack-sortable permutations) to validate and refine results.
• Proposes a general framework for counting decreasing plane trees with postorder readings in a given permutation class, generalizing the stack-sorting preimage problem.

Experimental results

Research questions

• RQ1 Can we compute $|s^{-1}( au)|$ efficiently for any permutation $\tau$, as posed by Bousquet-Mélou?
• RQ2 What is the exact enumeration of $s^{-1}(\text{Av}(132, 312))$ and $s^{-1}(\text{Av}(132, 231))$, and do they have the same generating function?
• RQ3 Is the sequence $|s^{-1}(\text{Av}_n(123\cdots m))|$ unimodal for $m \geq 2$?
• RQ4 Can we find a closed form for the number of permutations in $s^{-1}(\theta_{n,k})$ with $m$ peaks, where $\theta_{n,k}$ is a specific permutation with a decreasing prefix?
• RQ5 What is the exact enumeration of $s^{-1}(\text{Av}(231, 321))$, and does it match OEIS sequence A165543?

Key findings

• The paper provides a closed-form formula for $|s^{-1}(\text{Av}_n(132, 312))|$ and $|s^{-1}(\text{Av}_n(132, 231))|$, both with generating function $\frac{1 - 2x - \sqrt{1 - 4x - 4x^2}}{4x}$.
• It proves that $|s^{-1}(\text{Av}_n(123\cdots m))|$ is unimodal for $m \geq 2$, based on initial terms computed for $m=4$.
• For the permutation $\theta_{n,k} = (k+1)k(k-1)\cdots 321(k+2)(k+3)\cdots n$, the number of preimages with $m$ peaks is given by a sum over compositions involving $V(q_t, j_t)$, a function derived from valid hook configurations.
• The generating function for $s^{-1}(\text{Av}(231, 321))$ is conjectured to be $\frac{1}{1 - xC(xC(x))}$, where $C(x)$ is the Catalan generating function.
• The paper resolves a problem posed by Bruner by enumerating $s^{-1}(\text{Av}(132, 312))$ and providing a generating function for its preimages.
• It establishes that preimage sets $s^{-1}(\text{Av}(\tau^{(1)}, \dots, \tau^{(r)}))$ often form permutation classes themselves, enabling new combinatorial interpretations and identities.