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Inverse descent statistic for André and simsun permutations

Guo-Niu Han, Kathy Q. Ji, Huan Xiong

TL;DR

The paper addresses the equidistribution of the inverse-descent statistic together with the descent and major index across three Euler-number permutation models: André I, André II, and simsun permutations. It develops a tree-structure approach by mapping permutations to increasing binary trees, and shows that descent-related statistics are determined by tree shape, enabling a reduction to shape-level equidistribution. The authors establish two key shape-level results: (a) ides is equidistributed between André I and André II with the same tree shape, and (b) ides is equidistributed between André II and simsun permutations with the same tree shape, the latter via a bijection that preserves and relates several statistics. A central technical device is a set of refinements of Stanley's Shuffle Theorem to handle refined shuffle constructions, which underpin the equidistribution and yield explicit relations among inv, imaj, and RLmin under the constructed bijections. The findings thus extend Euler-number combinatorics by proving a robust tri-variate equidistribution and linking shuffle-compatibility with tree-shape properties.

Abstract

Simsun permutations, André I permutations and André II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of André I permutations and the set of simsun permutations. In this paper, we prove that the trivariate statistic (ides, des, maj), comprising the inverse descent, descent, and major index, are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of André I permutations and the set of André II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.

Inverse descent statistic for André and simsun permutations

TL;DR

The paper addresses the equidistribution of the inverse-descent statistic together with the descent and major index across three Euler-number permutation models: André I, André II, and simsun permutations. It develops a tree-structure approach by mapping permutations to increasing binary trees, and shows that descent-related statistics are determined by tree shape, enabling a reduction to shape-level equidistribution. The authors establish two key shape-level results: (a) ides is equidistributed between André I and André II with the same tree shape, and (b) ides is equidistributed between André II and simsun permutations with the same tree shape, the latter via a bijection that preserves and relates several statistics. A central technical device is a set of refinements of Stanley's Shuffle Theorem to handle refined shuffle constructions, which underpin the equidistribution and yield explicit relations among inv, imaj, and RLmin under the constructed bijections. The findings thus extend Euler-number combinatorics by proving a robust tri-variate equidistribution and linking shuffle-compatibility with tree-shape properties.

Abstract

Simsun permutations, André I permutations and André II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of André I permutations and the set of simsun permutations. In this paper, we prove that the trivariate statistic (ides, des, maj), comprising the inverse descent, descent, and major index, are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of André I permutations and the set of André II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.

Paper Structure

This paper contains 4 sections, 17 theorems, 72 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Increasing binary trees
  3. A more general result on permutations
  4. simsun permutations and André II permutations

Key Result

Theorem 1

The trivariate statistic ($\mathop{\mathrm{ides}}\nolimits$, $\mathop{\mathrm{des}}\nolimits$, $\mathop{\mathrm{maj}}\nolimits$ ) are equidistributed over the set of $n$-André I permutations, $n$-André II permutations and $n$-simsun permutations, i.e.,

Figures (6)

  • Figure 1: An inductive definition of $T_\pi$
  • Figure 2: The bijection between 4-André I trees and 4-André I permutations
  • Figure 3: The bijection between 4-André II trees and 4-André II permutations
  • Figure 4: The bijection between 4-simsun trees and 4-simsun permutations
  • Figure : (a) $8$-André I permutations:
  • ...and 1 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Definition 3: The map $\Psi$
  • Proposition 4: chow2011countingFH01
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • Corollary 8
  • Theorem 9: Stanley's Shuffle Theorem
  • Proposition 10
  • ...and 13 more