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Journal of the London Mathematical Society Advance Access originally published online on March 7, 2008
Journal of the London Mathematical Society 2008 77(3):627-646; doi:10.1112/jlms/jdn004
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© 2008 London Mathematical Society

Bimahonian distributions

Hélène Barcelo

Department of Mathematics and Statistics
Arizona State University
Tempe, AZ 85287
USA
barcelo@asu.edu

Victor Reiner and Dennis Stanton

School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA
reiner@math.umn.edu

Motivated by permutation statistics, we define, for any complex reflection group W, a family of bivariate generating functions W{sigma}(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{sigma}(t, q) satisfies a ‘bicyclic sieving phenomenon’ which combinatorially interprets its values when t and q are certain roots of unity.

The first author is supported by NSA grant H98230 [GenBank] -05-1-0256. The second and third authors are supported by NSF grants DMS-0601010 and DMS-0503660, respectively.

2000 Mathematics Subject Classification 05E10, 20F55, 13A50.

Received May 15, 2007; revised November 27, 2007; published online March 7, 2008.


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