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Pattern avoidance in nonnesting permutations

Sergi Elizalde, Amya Luo

TL;DR

This work investigates pattern avoidance in nonnesting permutations of $[n]\sqcup[n]$, focusing on avoiding sets of length-3 patterns and several length-4 patterns. By leveraging decompositions, recurrences, and bijections, the authors derive closed formulas and generating functions, revealing Catalan- and Fibonacci-number phenomena in numerous cases. They show how symmetry (reversal/complementation) reduces many problems to a few core cases, and they connect pattern-avoidance counts to classical combinatorial objects like Dyck and grand-Dyck words. The results highlight both similarities and key differences with the corresponding noncrossing setting and lay groundwork for future exploration of darker corners of pattern-avoidance in structured matchings and related hyperplane arrangements.

Abstract

Nonnesting permutations are permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid subsequences of the form $abba$ for any $a\neq b$. These permutations have recently been studied in connection to noncrossing (also called quasi-Stirling) permutations, which are those that avoid subsequences of the form $abab$, and in turn generalize the well-known Stirling permutations. Inspired by the work by Archer et al. on pattern avoidance in noncrossing permutations, we consider the analogous problem in the nonnesting case. We enumerate nonnesting permutations that avoid each set of two or more patterns of length 3, as well as those that avoid some sets of patterns of length 4. We obtain closed formulas and generating functions, some of which involve unexpected appearances of the Catalan and Fibonacci numbers. Our proofs rely on decompositions, recurrences, and bijections.

Pattern avoidance in nonnesting permutations

TL;DR

This work investigates pattern avoidance in nonnesting permutations of , focusing on avoiding sets of length-3 patterns and several length-4 patterns. By leveraging decompositions, recurrences, and bijections, the authors derive closed formulas and generating functions, revealing Catalan- and Fibonacci-number phenomena in numerous cases. They show how symmetry (reversal/complementation) reduces many problems to a few core cases, and they connect pattern-avoidance counts to classical combinatorial objects like Dyck and grand-Dyck words. The results highlight both similarities and key differences with the corresponding noncrossing setting and lay groundwork for future exploration of darker corners of pattern-avoidance in structured matchings and related hyperplane arrangements.

Abstract

Nonnesting permutations are permutations of the multiset that avoid subsequences of the form for any . These permutations have recently been studied in connection to noncrossing (also called quasi-Stirling) permutations, which are those that avoid subsequences of the form , and in turn generalize the well-known Stirling permutations. Inspired by the work by Archer et al. on pattern avoidance in noncrossing permutations, we consider the analogous problem in the nonnesting case. We enumerate nonnesting permutations that avoid each set of two or more patterns of length 3, as well as those that avoid some sets of patterns of length 4. We obtain closed formulas and generating functions, some of which involve unexpected appearances of the Catalan and Fibonacci numbers. Our proofs rely on decompositions, recurrences, and bijections.

Paper Structure

This paper contains 11 sections, 53 theorems, 110 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Patterns of length 3
  3. Avoiding one pattern
  4. Avoiding two patterns
  5. Avoiding three patterns
  6. Avoiding four or five patterns
  7. Some patterns of length 4
  8. Patterns whose repeated letters are in the middle
  9. Other patterns whose repeated letters are adjacent
  10. Patterns with non-adjacent repeated letters
  11. Further research

Key Result

Theorem 2.1

For all $n\ge1$, we have $c_n(112)=C_n$.

Figures (4)

  • Figure 1: The permutation $1521352434$ is nonnesting, but the permutation $13241342$ is not.
  • Figure 2: The subsets of $\mathcal{C}_n(1223,1332,2331)$ enumerated in this section, along with the theorem number.
  • Figure 3: The subsets of $\mathcal{C}_n(1332,2113,2331)$ enumerated in this section.
  • Figure 4: The subsets of $\mathcal{C}_n(1223,2331,3112)$ enumerated in this section.

Theorems & Definitions (105)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 95 more