On the product of generating functions for domino and bi-tableaux

TL;DR

This paper explores the generating functions of domino and bi-tableaux, deriving new formulas for type B Littlewood-Richardson coefficients using quasisymmetric functions, extending classical results for Young tableaux.

Contribution

It introduces novel formulas for type B Littlewood-Richardson coefficients involving bi-tableaux and domino tableaux through weak composition and Chow's quasisymmetric functions.

Findings

New formulas for type B Littlewood-Richardson coefficients

Connections between generating functions and quasisymmetric functions

Extensions of classical Young tableaux results

Abstract

The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak composition quasisymmetric functions and Chow's quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableaux. We look at their generalisations involving a set of bi-tableaux and domino tableaux.