Matrices, Characters and Descents

TL;DR

This paper introduces a new family of asymmetric Walsh-Hadamard type matrices, explores their properties including determinants and eigenvalues, and applies these findings to derive combinatorial enumeration formulas involving symmetric group characters.

Contribution

It presents a novel class of matrices and demonstrates their application in deriving character-based formulas for counting combinatorial objects with specific descent sets.

Findings

Computed determinants and eigenvalues of the new matrices

Established invertibility leading to character formulas

Derived expressions for counting sets with prescribed descent sets

Abstract

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain character formulas are invertible, yielding expressions for the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group.