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Sorting inversion sequences

Toufik Mansour, Howard Skogman, Rebecca Smith

TL;DR

This work studies pattern-avoidance in inversion sequences under sorting machines, focusing on stack- and pop-stack-based sorting. It establishes that stack-sortable inversion sequences correspond to $120$-avoiding words and develops a generating-tree framework to classify and enumerate sortable classes, including exact generating functions for several key pattern-avoiding families. It also extends to depth-restricted and generalized pop-stack models, deriving precise avoidance criteria (e.g., $120,\,201,\,1010$) and providing explicit trees and initial coefficients that illuminate the structure of sortable sequences. By connecting to layered permutations and known enumerative sequences (such as Catalan and Fibonacci numbers), the paper broadens the combinatorial understanding of sorting with restricted data structures and highlights several open directions for further enumeration and generalization.

Abstract

We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and generalizations, some of which we apply here. We give the classification of several classes of sortable inversion sequences in terms of pattern avoidance. We are able to provide an exact enumeration of some of the sortable classes in question using both classical approaches and a more recent strategy utilizing generating trees.

Sorting inversion sequences

TL;DR

This work studies pattern-avoidance in inversion sequences under sorting machines, focusing on stack- and pop-stack-based sorting. It establishes that stack-sortable inversion sequences correspond to -avoiding words and develops a generating-tree framework to classify and enumerate sortable classes, including exact generating functions for several key pattern-avoiding families. It also extends to depth-restricted and generalized pop-stack models, deriving precise avoidance criteria (e.g., ) and providing explicit trees and initial coefficients that illuminate the structure of sortable sequences. By connecting to layered permutations and known enumerative sequences (such as Catalan and Fibonacci numbers), the paper broadens the combinatorial understanding of sorting with restricted data structures and highlights several open directions for further enumeration and generalization.

Abstract

We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and generalizations, some of which we apply here. We give the classification of several classes of sortable inversion sequences in terms of pattern avoidance. We are able to provide an exact enumeration of some of the sortable classes in question using both classical approaches and a more recent strategy utilizing generating trees.
Paper Structure (10 sections, 26 theorems, 20 equations, 3 figures, 3 algorithms)

This paper contains 10 sections, 26 theorems, 20 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Stack sorting inversion sequences
  3. Pop stack sorting inversion sequences
  4. Restricting pop stack depth
  5. Generalizing pop stacks
  6. The generating tree method
  7. A formula for the generating function for the number of inversion sequences of length $n$ that avoid $120,201,210$
  8. The generating tree for the pop stack sortable inversion sequences
  9. Inversion sequences sortable by a pop stack of depth two
  10. Open Problems

Key Result

Proposition 2.1

The words that are stack sortable are exactly the words that avoid $120$.

Figures (3)

  • Figure 1: Moving to the state where $y$ will be forced to exit the pop stack before $x$ enters
  • Figure 2: The layered permutation $\pi=543216987$ and the layered word $w=211033275$.
  • Figure 3: A $(3,1)$-stack shown at several stages.

Theorems & Definitions (45)

  • Example 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 35 more