On a balanced property of compositions

Abstract

Let $S$ be a finite set of positive integers with largest element $m$. Let us randomly select a composition $a$ of the integer $n$ with parts in $S$, and let $m(a)$ be the multiplicity of $m$ as a part of $a$. Let $0\le r<q$ be integers, with $q\ge 2$, and let $p_{n,r}$ be the probability that $m(a)$ is congruent to $r\mathrm{mod}q$. We show that if $S$ satisfies a certain simple condition, then $\underset{n\to \mathrm{\infty }}{lim}{p}_{n,r}=1/q$. In fact, we show that an obvious necessary condition on $S$ turns out to be sufficient.