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.

We completely characterize the appearance of Schur functions corresponding to partitions of the form $\nu = (1^a, b)$ (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, $$s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu.$$ Specifically, we show that no Schur functions corresponding to hook shapes occur unless $\lambda$ and $\mu$ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm $s_{(1^c, d)}[s_{(1^a, b)}]$. We also consider the problem of adding a row or column so that $\nu$ is of the form $(1^a,b,c)$ or $(1^a, 2^b, c)$. This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.

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