Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 634823, 20 pages
http://dx.doi.org/10.1155/2013/634823
Research Article

Generalized Pattern-Matching Conditions for

1Department of Computer and Information Sciences, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK
2Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
3Department of Mathematics, University of Wisconsin, Eau Claire, WI 54702-4004, USA

Received 3 May 2012; Accepted 1 July 2012

Academic Editors: S. D. Georgiou and J. Siemons

Copyright © 2013 Sergey Kitaev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.