Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

We continue the study of the generalized pattern avoidance condition for , the wreath product of the cyclic group with the symmetric group , initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers. 1. Introduction The goal of this paper is to continue the study of pattern-matching conditions on the wreath product of the cyclic group and the symmetric group initiated in [1]. is the group of signed permutations where there are signs, , , , , , where is a 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 . Moreover, we think of the elements of as the colors of the corresponding elements of the underlying permutation . We define a concept for matchings in words over a finite alphabet . Given a word , let be the word found by replacing the largest integer that appears in by . For example, if , then . Given a word such that , we say that 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 . We say that avoids if there are no occurrences of in . There are a number of papers on pattern matching and pattern avoidance in [2–5]. We now present a selection of the previously studied definitions for pattern matching and avoidance, so that the reader may see how ours differs from and/or extends those in the literature. For example, the following pattern matching condition was studied in [3–5]. 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 (resp., . That is, an exact occurrence or an exact match of in an element is just an ordinary occurrence or match of in where the corresponding signs agree exactly. For example, Mansour [4] proved via recursion that, for any , the number of elements in which avoid exact occurrences of is . This generalized a result of Simion [6] who proved the same result for the hyperoctahedral group . Similarly, Mansour and West [5] determined the number of permutations in that avoid all possible exact occurrences of 2 or 3 element sets of