Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers

TL;DR

This paper proves conjectures about noncommutative symmetric functions, showing that certain transition matrices have nonnegative integer coefficients, using combinatorial statistics related to Genocchi numbers.

Contribution

It introduces two new composition-valued permutation statistics that generalize Genocchi number combinatorics and proves nonnegativity of transition matrices between bases.

Findings

Transition matrices have nonnegative integer coefficients.

Two new permutation statistics generalize Genocchi number combinatorics.

Confirmed conjectures on noncommutative symmetric functions.

Abstract

We prove conjectures of the third author [L. Tevlin, Proc. FPSAC'07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.