The Worpitzky identity for the groups of signed and even-signed permutations

The well-known Worpitzky identity provides a connection between two bases of : The standard basis  and the binomial basis , where the Eulerian numbers for the Coxeter group of type  (the symmetric group) serve as the entries of the transformation matrix.

Brenti has generalized this identity to the Coxeter groups of types  and  (signed and even-signed permutations groups, respectively) using generatingfunctionology.  

Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for its -analogue in the Coxeter groups of types  and . Our proofs utilize the language of -partitions for the  and  posets introduced by Chow and Stembridge, respectively. 

Joint work with David Garber and Mordechai Novick.