Bijective evaluation of the connection coefficients of the double coset algebra

TL;DR

This paper explicitly evaluates the generating series of connection coefficients for double cosets of the hyperoctahedral group, linking algebraic structures to combinatorial objects like hypermaps and forests.

Contribution

It provides a new explicit evaluation for the series when =(n), using combinatorial interpretations and bijections involving hypermaps and permuted forests.

Findings

Explicit formula for the series when =(n)

New bijective construction between hypermaps and forests

Connection between algebraic coefficients and combinatorial structures

Abstract

This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $\nu$, gives the spectral distribution of some random real matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $\nu=(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between locally orientable partitioned hypermaps and some permuted forests.