Generalized Pattern-Matching Conditions for Ck≀Sn

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product Ck≀Sn of the cyclic group Ck and the symmetric group Sn. In particular, we derive the generating functions for the number of matches that occur in elements of Ck≀Sn for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of Ck≀Sn. Our research leads to connections to many known objects/structures yet to be explained combinatorially.