On some colored Eulerian quasisymmetric functions

TL;DR

This paper explores colored Eulerian quasisymmetric functions, deriving generating functions, proving symmetric identities, confirming unimodality conjectures, and introducing new permutation statistics and formulas.

Contribution

It provides new generating function formulas, symmetric identities, and permutation statistics for colored Eulerian quasisymmetric functions, extending previous combinatorial and algebraic results.

Findings

Derived generating function formula from the Decrease value theorem.

Proved symmetric function generalizations of the Eulerian identity.

Confirmed unimodality conjecture for flag excedances over type B derangements.

Abstract

Recently, Hyatt introduced some colored Eulerian quasisymmetric function to study the joint distribution of excedance number and major index on colored permutation groups. We show how Hyatt's generating function formula for the fixed point colored Eulerian quasisymmetric functions can be deduced from the Decrease value theorem of Foata and Han. Using this generating function formula, we prove two symmetric function generalizations of the Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian quasisymmetric functions, which are specialized to the flag excedance numbers. Combinatorial proofs of those symmetrical identities are also constructed. We also study some other properties of the flag Eulerian quasisymmetric functions. In particular, we confirm a recent conjecture of Mongelli [Journal of Combinatorial Theory, Series A, 120 (2013) 1216--1234] about the unimodality of the generating function of the flag excedances over the type B derangements. Moreover, colored versions of the hook factorization and admissible inversions of permutations are found, as well as a new recurrence formula for the $(\maj-\exc,\fexc)$-$q$-Eulerian polynomials. We introduce a colored analog of Rawlings major index on colored permutations and obtain an interpretation of the colored Eulerian quasisymmetric functions as sums of some fundamental quasisymmetric functions related with them, by applying Stanley's $P$-partition theory and a decomposition of the Chromatic quasisymmetric functions due to Shareshian and Wachs.