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(modm). 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.

Source : oai:HAL:hal-01179220v1

Volume: Vol. 16 no. 1 (in progress)

Section: Combinatorics

Published on: May 22, 2014

Submitted on: July 13, 2013

Keywords: discrete mathematics, combinatorics,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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