Bijective evaluation of the connection coefficients of the double coset algebra

International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some ra...

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Bibliographic Details
Published in:Discrete Mathematics & Theoretical Computer Science
Main Authors: Morales, Alejandro H., Vassilieva, Ekaterina A.
Other Authors: Department of Mathematics MIT, Massachusetts Institute of Technology (MIT), Laboratoire d'informatique de l'École polytechnique Palaiseau (LIX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Bousquet-Mélou, Mireille and Wachs, Michelle and Hultman, Axel
Format: Conference Object
Language:English
Published: HAL CCSD 2011
Subjects:
double coset algebra
connection coefficients
locally orientable hypermaps
forests
[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Iceland
Online Access:https://hal.inria.fr/hal-00755301
https://hal.inria.fr/hal-00755301v2/document
https://hal.inria.fr/hal-00755301v2/file/dmAO0160.pdf
https://doi.org/10.46298/dmtcs.2944
Description
Summary:International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests. Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

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