Restricted Stacks as Functions

TL;DR

This paper generalizes the stack sort algorithm by avoiding sets of permutations, classifies when the associated maps are bijective, and explores their combinatorial properties, including preimages and periodic points.

Contribution

It classifies when the generalized stack sort maps are bijective and characterizes the preimages, extending understanding of permutation avoidance in stack sorting.

Findings

The map $s_T$ is bijective for specific sets $T$.

Preimages under $s_T$ are bounded by Catalan numbers.

Exact conditions when the bound is sharp for single-element sets.

Abstract

The stack sort algorithm has been the subject of extensive study over the years. In this paper we explore a generalized version of this algorithm where instead of avoiding a single decrease, the stack avoids a set $T$ of permutations. We let $s_T$ denote this map. We classify for which sets $T$ the map $s_T$ is bijective. A corollary to this answers a question of Baril, Cerbai, Khalil, and Vajnovszki about stack sort composed with $s_{\{\sigma,\tau\}}$, known as the $(\sigma,\tau)$-machine. This fully classifies for which $\sigma$ and $\tau$ the preimage of the identity under the $(\sigma,\tau)$-machine is counted by the Catalan numbers. We also prove that the number of preimages of a permutation under the map $s_T$ is bounded by the Catalan numbers, with a shift of indices. For $T$ of size 1, we classify exactly when this bound is sharp. We also explore the periodic points and maximum number of preimages of various $s_T$ for $T$ containing two length $3$ permutations.