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On the joint distributions of succession and Eulerian statistics

Shi-Mei Ma, Hao Qi, Jean Yeh, Yeong-Nan Yeh

Abstract

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. As an generalization a result of Diaconis-Evans-Graham (Adv. in Appl. Math., 61 (2014), 102-124), we show that two triple set-valued statistics of permutations are equidistributed on symmetric groups. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc (p,q)-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomials.

On the joint distributions of succession and Eulerian statistics

Abstract

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. As an generalization a result of Diaconis-Evans-Graham (Adv. in Appl. Math., 61 (2014), 102-124), we show that two triple set-valued statistics of permutations are equidistributed on symmetric groups. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc (p,q)-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomials.
Paper Structure (13 sections, 23 theorems, 118 equations, 1 figure)

This paper contains 13 sections, 23 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Triple and quadruple statistics
  4. Main results
  5. Proof of Theorem \ref{['mainthm01']}
  6. Proof of Theorem \ref{['NewEu']}
  7. Another interpretation of the coefficients $\gamma_{n,i,j}$
  8. Proper left-to-right minimum statistic
  9. Relationship to ${\rm fix\,}$ and ${\rm cyc\,}$ (p,q)-Eulerian polynomials
  10. A fundamental lemma on (p,q)-Eulerian polynomials
  11. Four-variable polynomials
  12. Five-variable polynomials
  13. Six-variable polynomials

Key Result

Proposition 1

The polynomials $P_n(x)$ are bi-$\gamma$-positive.

Figures (1)

  • Figure 1: The labeling of a 0-1-2 increasing planted tree on $\{0,1,2,\ldots,8\}$.

Theorems & Definitions (38)

  • Proposition 1
  • Definition 2
  • Proposition 3
  • Corollary 4
  • Proposition 5
  • Definition 6
  • Theorem 7
  • Corollary 8
  • Theorem 9
  • Lemma 10
  • ...and 28 more