Enumeration of m-tuples of permutations and a new class of power bases for the space of symmetric functions

Abstract

In this paper, we continue a long line of research which shows that many generating function identities for various permutation statistics arise from well known symmetric function identities by applying certain ring homomorphisms on the ring of symmetric functions. This idea was first introduced in a 1993 paper of Brenti who used it to show that the generating functions of permutations of the symmetric group S n by descents and excedances could be derived in such a manner. In this paper, we define certain ( q , t )-analogues of Brenti's homomorphism that lead to generating functions for statistics on m-tuples of permutations. Our results generalize previous work of generating functions for permutations statistics due to Carlitz, Stanley, and Fedou and Rawlings. We also introduce some new bases of symmetric functions which are necessary to extend Brenti's results on generating functions for permutations by excedances to m-tuples of permutations. Finally, we study the image of our homomorphisms on analogues of the q-basis of symmetric functions studied by Ram and King and Wybourne to describe the irreducible characters of the Hecke algebras of type A.