Congruences of concave composition functions

Concave compositions are ordered partitions whose parts are decreasing towards a central part. We study the distribution modulo $a$ of the number of concave compositions. Let $c(n)$ be the number of concave compositions of $n$ having even length. It is easy to see that $c(n)$ is even for all $n\geq1$. Refining this fact, we prove that $\#\{n<X:c(n)\equiv 0\pmod 4\}\gg\sqrt{X}$ and also that for every $a>2$ and at least two distinct values of $r\in\{0,1,\dotsc,a-1\}$, $\#\{n<X: c(n)\equiv r\pmod{a}\} > \frac{\log_2\log_3 X}{a}.$ We obtain similar results for concave compositions of odd length.

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