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An insertion process and a parity based equidistribution

Umesh Shankar

Abstract

A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates descents with odd descent tops, $S_{12}$ enumerates odd-odd adjacent pairs, and $S_{17}$ records the largest integer $i$ such that $1, 2, \dots, i$ appear in left-to-right order. In this note, we resolve this conjecture affirmatively by providing a bijective proof. We introduce an insertion process that constructs a recursive involution on $\mathfrak{S}_n$ that swaps $S_{10}$ and $S_{12}$ while keeping $S_{17}$ unchanged.

An insertion process and a parity based equidistribution

Abstract

A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics and are equidistributed over the symmetric group . Here, enumerates descents with odd descent tops, enumerates odd-odd adjacent pairs, and records the largest integer such that appear in left-to-right order. In this note, we resolve this conjecture affirmatively by providing a bijective proof. We introduce an insertion process that constructs a recursive involution on that swaps and while keeping unchanged.
Paper Structure (3 sections, 4 theorems, 1 equation, 3 tables)

This paper contains 3 sections, 4 theorems, 1 equation, 3 tables.

Table of Contents

  1. Introduction
  2. The involution
  3. Examples

Key Result

Theorem 1

The quadruples $(T_1,T_2,T_3,S_{17})$ and $(T_2,T_1,T_3,S_{17})$ are equidistributed over $\mathfrak{S}_n$ for all $n\in \mathbb{N}$.

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm: main']}
  • proof : Proof of Corollary \ref{['cor: S10_S12']}
  • Example 5
  • Example 6
  • ...and 1 more