2006/03/13 by S. Heubach, Heubach, S., T. Mansour +1
Mathematics · #05A05 (primary) #05A15 #05A16 (secondary) #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A05 #msc:05A15 #msc:05A16
paper · pdf · doi:10.48550/arxiv.math/0603285
20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer Conference
arxiv created 2006/03/13 · arxiv updated 2009/12/01
Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.