Direct bijective computation of the generating series for 2 and 3-connection coefficients of the symmetric group

TL;DR

This paper introduces a new bijective method to explicitly compute the generating series for 2 and 3-connection coefficients of the symmetric group, linking algebraic and combinatorial approaches.

Contribution

It provides a novel bijective approach and explicit formulas for the generating series of these connection coefficients, extending previous algebraic methods.

Findings

Explicit closed-form evaluation of generating series for connection coefficients.

A new bijection involving modified oriented tricolored trees.

Reduction of the bijection to factorizations into two permutations.

Abstract

We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation in terms of unicellular constellations on orientable surfaces. Algebraic computation of these coefficients was first done by Jackson using irreducible characters of the symmetric group. However, bijective computations of these coefficients are so far limited to very special cases. Thanks to a new bijection that refines the work of Schaeffer and Vassilieva, and Vassilieva, we give an explicit closed form evaluation of the generating series for these coefficients. The main ingredient in the bijection is a modified oriented tricolored tree tractable to enumerate. Finally, reducing this bijection to factorizations of a long cycle into two permutations, we get the analogue formula for the corresponding generating series.