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Bijective evaluation of the connection coefficients of the double coset algebra

Abstract : 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.
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Alejandro H. Morales, Ekaterina A. Vassilieva. Bijective evaluation of the connection coefficients of the double coset algebra. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.681-692, ⟨10.46298/dmtcs.2944⟩. ⟨hal-00755301v2⟩

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