New pattern matching conditions for wreath products of the cyclic groups with symmetric groups

TL;DR

This paper introduces new pattern matching conditions for wreath products of cyclic and symmetric groups, providing generating functions for match counts, non-overlapping matches, and elements with exactly two matches, expanding combinatorial understanding.

Contribution

It develops a novel pattern matching condition involving signs and permutations, and derives generating functions for match distributions in wreath products.

Findings

Generated functions for pattern matches of length 2

Derived distributions for non-overlapping matches

Counted elements with exactly two non-overlapping matches

Abstract

We present several multi-variable generating functions for a new pattern matching condition on the wreath product of the cyclic group and the symmetric group. Our new pattern matching condition requires that the underlying permutations match in the usual sense of pattern matching for the symmetric group 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 the wreath product 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. We also provide multi-variable generating functions for the distribution of non-overlapping matches and for the number of elements of the wreath product which have exactly 2 matches which do not overlap for several patterns of length 2.