Cointeraction on noncrossing partitions and related polynomial invariants

TL;DR

This paper explores the algebraic structure of noncrossing partitions in free probability, introducing a polynomial invariant with combinatorial and algebraic significance, and revealing connections to various mathematical objects.

Contribution

It introduces a new polynomial invariant for noncrossing partitions that respects a dual coproduct structure and links to multiple combinatorial and algebraic concepts.

Findings

The polynomial invariant respects the product and both coproducts.

Explicit combinatorial interpretation and evaluation at -1.

Connections to harmonic nested sums, Riordan arrays, and Stirling numbers.

Abstract

We study the structure of two cointeracting bialgebras on noncrossing partitions appearing in the theory of free probability. The first coproduct is given by separation of the blocks of the partitions into two parts, with respect to the nestings, while the second one is given by fusion of blocks. This structure implies the existence of a unique polynomial invariant respecting the product and both coproducts. We give a combinatorial interpretation of this invariant, study its values at -1 and use it for the computation of the antipode. We also give several results on its coefficients when applied to noncrossing partitions with no nesting. This leads to unexpected links with harmonic nested sums, Riordan arrays, composition of formal series and generalized Stirling numbers. This polynomial invariant is shown to be related to other ones, counting increasing or strictly increasing maps for the nesting order on noncrossing partitions, through the action of several characters.