Matrices, Characters and Descents
Ron M. Adin, Yuval Roichman

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.
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