Generalized Pattern-Matching Conditions for

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of 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 . Our research leads to connections to many known objects/structures yet to be explained combinatorially. 1. Introduction The goal of this paper is to study pattern-matching conditions on the wreath product of the cyclic group and the symmetric group . is the group of signed permutations where we allow signs of the form for some primitive th root of unity . We can think of the elements as pairs where and . For ease of notation, if where for , then we simply write where . Given a sequence of distinct integers, let red be the permutation found by replacing the th largest integer that appears in by . For example, if , then . Given a permutation in the symmetric group , we say a permutation has a -match starting at position provided . Let -mch be the number of -matches in the permutation . Similarly, we say that occurs in if there exist such that red . We say that ？？ avoids if there are no occurrences of in . We can define similar notions for words over a finite alphabet . Given a word , let red be the word found by replacing the largest integer that appears in by . For example, if , then red . Given a word such that red , we say a word has a -match starting at position provided . Let -mch be the number of -matches in the word . Similarly, we say that occurs in a word if there exist such that red . We say that ？？ avoids if there are no occurrences of in . There are a number of papers on pattern matching and pattern avoidance in [1–4]. For example, the following pattern matching condition was studied in [2–4]. Definition 1. Let , be a subset of and . (1)One says that has an exact occurrence of (resp., ) if there are such that (resp., ).(2)One says that ？？avoids an exact occurrence of？？ (resp., ) if there are no exact occurrences of (resp., ) in .(3)One says that there is an exact？？ -match in？？ ？？ starting at position？？ (resp., exact？？ -match in？？ ？？starting at position？？ ) if That is, an exact occurrence or an exact match of in an element is just an ordinary occurrence or match of in where the