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On stable patterns and properties on permutations of multisets

Shaoshi Chen, Hanqian Fang, Sergey Kitaev

Abstract

In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as stable. We provide a complete characterization of stable classical patterns, showing that the only such patterns are those of length one or two. For consecutive patterns, we establish the stability of all monotone patterns and also identify a large class of unstable patterns. We conjecture that monotone patterns are the only stable consecutive patterns. All stability results in this paper are proven via explicit bijections, which provide new combinatorial interpretations of the symmetry of the generating functions. As an application, we use stability to derive recurrence relations for the ascent distribution on permutations of multisets, resulting in a generalization of Eulerian numbers.

On stable patterns and properties on permutations of multisets

Abstract

In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as stable. We provide a complete characterization of stable classical patterns, showing that the only such patterns are those of length one or two. For consecutive patterns, we establish the stability of all monotone patterns and also identify a large class of unstable patterns. We conjecture that monotone patterns are the only stable consecutive patterns. All stability results in this paper are proven via explicit bijections, which provide new combinatorial interpretations of the symmetry of the generating functions. As an application, we use stability to derive recurrence relations for the ascent distribution on permutations of multisets, resulting in a generalization of Eulerian numbers.
Paper Structure (6 sections, 16 theorems, 33 equations, 2 figures)

This paper contains 6 sections, 16 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability and classical patterns
  4. Stability and consecutive patterns
  5. Applications of the stability: generalized Eulerian numbers
  6. Conclusion

Key Result

Theorem 3.1

The length-$2$ classical patterns, i.e., the patterns $12$ and $21$, are stable.

Figures (2)

  • Figure 1: Steps for obtaining the extended permutation $\pi$ of the pattern $p=\underline{24135}$ at the index $4$.
  • Figure 2: The number $\delta_j^i$ denotes the number of points in each region.

Theorems & Definitions (37)

  • Example 2.3
  • Theorem 3.1
  • proof
  • Example 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Remark 3.6
  • Lemma 4.1
  • ...and 27 more