A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions

TL;DR

This paper introduces a new basis for quasi-symmetric functions and noncommutative symmetric functions by lifting cycle-index polynomials, providing new algebraic structures and combinatorial formulas.

Contribution

It defines a novel basis of quasi-symmetric functions via cycle-index polynomials and explores its dual noncommutative symmetric functions with explicit product and recurrence formulas.

Findings

New basis of quasi-symmetric functions constructed

Product formula and recurrence for the basis derived

Identification of noncommutative symmetric functions with a specific quotient

Abstract

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to $QSym$. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of FQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312 \equiv 132$.