q-Stirling numbers in type B

Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of typ...

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Bibliographic Details
Published inEuropean journal of combinatorics Vol. 118; p. 103899
Main Authors Sagan, Bruce E., Swanson, Joshua P.
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.05.2024
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ISSN0195-6698
1095-9971
DOI10.1016/j.ejc.2023.103899

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Summary:Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson–Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2023.103899