Congruence successions in compositions

TL;DR

This paper studies the enumeration of compositions based on the occurrence of adjacent parts with congruent values modulo m, providing formulas, special cases, and asymptotic estimates for compositions with no such successions.

Contribution

It introduces a general formula for counting compositions with a given number of m-congruence successions, extending previous results and analyzing the case m=2 in detail.

Findings

Derived a general counting formula for compositions with m-congruence successions.

Obtained specific enumerative results for the case m=2.

Provided an asymptotic estimate for compositions with no m-congruence successions.

Abstract

A \emph{composition} is a sequence of positive integers, called \emph{parts}, having a fixed sum. By an \emph{$m$-congruence succession}, we will mean a pair of adjacent parts $x$ and $y$ within a composition such that $x\equiv y(\text{mod} m)$. 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.