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Bimahonian distributions
Author(s) -
Barcelo Hélène,
Reiner Victor,
Stanton Dennis
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn004
Subject(s) - mathematics , permutation (music) , invariant (physics) , combinatorics , bivariate analysis , diagonal , pure mathematics , statistics , physics , geometry , mathematical physics , acoustics
Motivated by permutation statistics, we define, for any complex reflection group W , a family of bivariate generating functions W σ ( t, q ). They are defined either in terms of Hilbert series for W ‐invariant polynomials when W acts diagonally on two sets of variables or, equivalently, as sums involving the fake degrees of irreducible representations for W . It is shown that W σ ( t, q ) satisfies a ‘bicyclic sieving phenomenon’ which combinatorially interprets its values when t and q are certain roots of unity.