Inversion of some series of free quasi-symmetric functions

TL;DR

This paper provides a combinatorial formula for inverting certain sums of free quasi-symmetric functions, confirming conjectures in special cases related to compositions with specific part restrictions.

Contribution

It introduces a new combinatorial inversion formula for free quasi-symmetric functions, resolving three conjectured cases.

Findings

Derived explicit combinatorial formulas for inverses

Confirmed conjectures for specific composition restrictions

Enhanced understanding of free quasi-symmetric function structures

Abstract

We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form F_{\omega(I)} where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B. C. V. Ung in [Proc. FPSAC'98, Toronto].