Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 251 -- 266

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MULTINOMIAL FIX-MAHONIAN STATISTICS

Hery Randriamaro

Abstract. The permutation statistics fix, des, maj, and inv have different original contexts, and appear in diverse scientific domains such as probability, physics, and genomics. But so far, they only meet together in generating functions and equidistributions. Examples are the generating function of (inv,  des,  maj) computed by Garsia and Gessel, and the equidistributivity of (fix,  des,  maj) and (fix,  dez,  maz) proved by Foata and Han. Recently, Tsilevich and Vershik determined the eigenvalues and multiplicities of (des(σ τ-1))σ, τ ∈ 𝔖n, (maj(σ τ-1))σ, τ ∈ 𝔖n, and (inv(σ τ-1))σ, τ ∈ 𝔖n, and Tsilevich determined those of (fix(σ τ-1))σ, τ ∈ 𝔖n. This article studies combinations of these statistics in terms of matrices. For that, the regular representation of the sum over all permutations weighted by the sum of their multinomial descents, inversions, and fixed points is considered. We compute the eigenvalues and multiplicities of that matrix. Then, we deduce those of (des(σ τ-1) + maj(σ τ-1) + inv(σ τ-1) + fix(σ τ-1))σ, τ ∈ 𝔖n.

2020 Mathematics Subject Classification: 05A05; 05E10; 15A18
Keywords: Permutation statistics; Algebra representation; Matrix spectrum

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Hery Randriamaro
Universität Kassel,
Institut für Mathematik,
Heinrich-Plett-Straße 40, 34132 Kassel, Germany.
e-mail: hery.randriamaro@mathematik.uni-kassel.de
https://www.mathematik.uni-kassel.de/randriamaro/index.html

https://www.utgjiu.ro/math/sma