Multinomial fix-Mahonian statistics
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, d...
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University Constantin Brancusi of Targu-Jiu
2025-04-01
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| Series: | Surveys in Mathematics and its Applications |
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| Online Access: | https://www.utgjiu.ro/math/sma/v20/p20_13.pdf |
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| author | Hery Randriamaro |
| author_facet | Hery Randriamaro |
| author_sort | Hery Randriamaro |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-760ea843a8b54dc8a4a01d2ae8d7fd1a |
| institution | DOAJ |
| issn | 1843-7265 1842-6298 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | University Constantin Brancusi of Targu-Jiu |
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| series | Surveys in Mathematics and its Applications |
| spelling | doaj-art-760ea843a8b54dc8a4a01d2ae8d7fd1a2025-08-20T02:39:38ZengUniversity Constantin Brancusi of Targu-JiuSurveys in Mathematics and its Applications1843-72651842-62982025-04-0120 (2025)251266Multinomial fix-Mahonian statistics Hery Randriamaro0Universität Kassel, Institut für Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, GermanyThe 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. https://www.utgjiu.ro/math/sma/v20/p20_13.pdfpermutation statisticsalgebra representationmatrix spectrum |
| spellingShingle | Hery Randriamaro Multinomial fix-Mahonian statistics Surveys in Mathematics and its Applications permutation statistics algebra representation matrix spectrum |
| title | Multinomial fix-Mahonian statistics |
| title_full | Multinomial fix-Mahonian statistics |
| title_fullStr | Multinomial fix-Mahonian statistics |
| title_full_unstemmed | Multinomial fix-Mahonian statistics |
| title_short | Multinomial fix-Mahonian statistics |
| title_sort | multinomial fix mahonian statistics |
| topic | permutation statistics algebra representation matrix spectrum |
| url | https://www.utgjiu.ro/math/sma/v20/p20_13.pdf |
| work_keys_str_mv | AT heryrandriamaro multinomialfixmahonianstatistics |