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Generalized pattern matching conditions for Ck o Sn

Kitaev, Sergey and Niedermaier, Andrew and Remmel, Jeffrey and Riehl, Manda (2013) Generalized pattern matching conditions for Ck o Sn. ISRN Combinatorics, 2013.

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Abstract

We present several multi-variable generating functions for a new pattern matching condition on the wreath product Ck o Sn of the cyclic group Ck and the symmetric group Sn. Our new pattern matching condition requires that the underlying permutations match in the usual sense of pattern matching for Sn and that the corresponding sequence of signs match in the sense of words, rather than the exact equality of signs which has been previously studied. We produce the generating functions for the number of matches that occur in elements of Ck o Sn for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an in¯nite number of variables to simple symmetric function identities. We also provide multi-variable generating functions for the distribution of nonoverlapping matches and for the number of elements of Ck o Sn which have exactly 2 matches which do not overlap for several patterns of length 2.