2013/07/28 by Mansour, Toufik, Shattuck, Mark, Wilson, Mark C.
#05A15 #05A16 #05A19 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1307.7390
A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x≡ y(mod m). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.