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A descent-excedance correspondence in colored permutation groups

Hiranya Kishore Dey, Umesh Shankar, Sivaramakrishnan Sivasubramanian

Abstract

It is well known that descents and excedances are equidistributed in the symmetric group. We show that the descent and excedance enumerators, summed over permutations with a fixed first letter are identical when we perform a simple change of the first letter. We generalize this to type B and other colored permutation groups. We are led to defining descents and excedances through linear orders. With respect to a particular order, when the number of colors is even, we get a result that generalizes the type B results. Lastly, we get a type B counterpart of Conger's result which refines the well known Carlitz identity.

A descent-excedance correspondence in colored permutation groups

Abstract

It is well known that descents and excedances are equidistributed in the symmetric group. We show that the descent and excedance enumerators, summed over permutations with a fixed first letter are identical when we perform a simple change of the first letter. We generalize this to type B and other colored permutation groups. We are led to defining descents and excedances through linear orders. With respect to a particular order, when the number of colors is even, we get a result that generalizes the type B results. Lastly, we get a type B counterpart of Conger's result which refines the well known Carlitz identity.

Paper Structure

This paper contains 7 sections, 18 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorems \ref{['thm: ldes-lexc']} and \ref{['thm: criss-cross']}
  4. Colored permutations with an even number of colors
  5. Real-rootedness of the type B restricted Eulerian polynomials
  6. Gamma positivity of symmetrized type B Eulerian polynomials
  7. Proof of Theorem \ref{['thm: typeb-typea-des-carlitz']}

Key Result

Theorem 1

With the above notation, we have

Theorems & Definitions (44)

  • Theorem 1: Conger
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7: Nevo, Petersen and Tenner
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 34 more