Diagonalization of Fix-Mahonian Matrices

TL;DR

This paper analyzes the spectral properties of a matrix derived from permutation statistics, providing explicit eigenvalues and multiplicities, which unify various permutation statistics in a single framework.

Contribution

It computes the spectrum and multiplicities of a matrix associated with permutation statistics, offering a unified approach to multiple permutation measures.

Findings

Explicit spectrum and multiplicities of the matrix are derived.

The matrix encapsulates multiple permutation statistics simultaneously.

Results enable analysis of permutation statistics through spectral methods.

Abstract

Consider the regular representation of the sum over all permutations weighted by the sum of their descent, inversion, and fixed point multinomials. We compute the spectrum and the multiplicities of its elements of that matrix. Note that those multinomial statistics allow to apply the result on several permutation statistics like the number of fixed points, of descents, of inversions, and the major index at the same time.