Congruences of concave composition functions

TL;DR

This paper investigates the distribution of the number of concave compositions modulo various integers, revealing growth patterns and congruence properties for compositions of both even and odd lengths.

Contribution

It provides new results on the modular distribution of concave compositions, including lower bounds and density estimates for their congruence classes.

Findings

Number of even-length concave compositions divisible by 4 grows at least as fast as the square root of X.

For any modulus greater than 2, at least two residue classes have a lower bound proportional to log log X.

Similar distribution results are established for odd-length concave compositions.

Abstract

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.