Sorting with pattern-avoiding stacks: the $132$-machine

TL;DR

This paper characterizes and enumerates permutations sortable by a two-stack machine where the first stack avoids the pattern 132, connecting it to lattice paths and set partitions, and solving an open problem.

Contribution

It provides a complete characterization and enumeration of permutations sortable with a 132-avoiding first stack, addressing an open problem and introducing new combinatorial tools.

Findings

Enumeration of sortable permutations with 132-avoiding stack

Connections established with lattice paths and set partitions

New proofs for pattern-avoiding restricted growth functions

Abstract

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari [CCF]. These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed in [CCF]. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.