Combinatorial Proofs of Bivariate Generating Functions on Coxeter Groups

McMahon’s result that states the length and major index statistics are equidistributed on the symmetric group Sn has generalizations to other Coxeter groups. Adin, Brenti and Roichman have defined analogues of those statistics for the hyperoctahedral group, Bn and proved equidistribution theorems. Using combinatorial interpretations of Regev and Roichman’s statistics length and delent, we consider a specialization of the bimahonian generating function for the alternating group An. We define a new delent statistic, delL, for the alternating subgroup of the hyperoctahedral group, Ln ⊂ Bn and give a combinatorial proof of the bimahonian generating function ∑ π∈Ln q lL(π)tdelL(π). Mathematics Subject Classification: 20F55, 05A15