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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| Published in | European journal of combinatorics Vol. 118; p. 103899 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.05.2024
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| Online Access | Get full text |
| ISSN | 0195-6698 1095-9971 |
| DOI | 10.1016/j.ejc.2023.103899 |
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| Abstract | 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. |
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| AbstractList | 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. |
| ArticleNumber | 103899 |
| Author | Swanson, Joshua P. Sagan, Bruce E. |
| Author_xml | – sequence: 1 givenname: Bruce E. surname: Sagan fullname: Sagan, Bruce E. email: sagan@math.msu.edu organization: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA – sequence: 2 givenname: Joshua P. surname: Swanson fullname: Swanson, Joshua P. email: swansonj@usc.edu organization: Department of Mathematics, University of Southern California, Los Angeles, CA 90007, USA |
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