Bimahonian distributions

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.