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Pattern-avoiding stabilized-interval-free permutations

Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, Michael D. Weiner

Abstract

In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.

Pattern-avoiding stabilized-interval-free permutations

Abstract

In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.
Paper Structure (11 sections, 12 theorems, 55 equations, 2 figures, 2 tables)

This paper contains 11 sections, 12 theorems, 55 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Avoiding a pattern of size 3
  3. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(132)\sim \mathcal{S}^{ \mathsf{sif}}_n(213)\sim \mathcal{S}^{ \mathsf{sif}}_n(321)$
  4. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(123)$
  5. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(231)\sim \mathcal{S}^{ \mathsf{sif}}_n(312)$
  6. Avoiding a pair of patterns of size 3
  7. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(132,231)\sim \mathcal{S}^{ \mathsf{sif}}_n(213,312)$
  8. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(123,132)\sim \mathcal{S}^{ \mathsf{sif}}_n(123,213)$
  9. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(132,213)$
  10. Enumeration of $\mathcal{S}^{ \mathsf{sif}}_n(123,231)\sim \mathcal{S}^{ \mathsf{sif}}_n(123,312)$
  11. Appendix

Key Result

Theorem 2.1

For $n\ge 1$, we have $\mathcal{S}^{ \mathsf{sif}}_n(321) = \mathcal{S}^{ \mathsf{ind}}_n(321)$. Therefore, $\lvert\mathcal{S}^{ \mathsf{sif}}_n(321)\rvert = C_{n-1}$.

Figures (2)

  • Figure 1: Construction $562134 \mapsto 67{\color{red1}3}2145 \mapsto 89{\color{red1}453}2167$.
  • Figure 2: Ordered forest corresponding to $\sigma=312495876$.

Theorems & Definitions (27)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark
  • Proposition 2.3: Eli04
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 17 more