凹复合函数的同余
Congruences of concave composition functions
摘要：
凹组合物是有序的分区，其部分朝向中心部分减少。研究了凹复合数的分布模$a$。令$c(n)$为具有偶数长度的$n$的凹组合的数目。很容易看出，$c(n)$甚至适用于所有$n\geq1$。精炼这一事实，我们证明，$$\#\{n<X:c(n)\equiv 0\pmod 4\}\gg\sqrt{X}$$，也为每个$a>2$和至少两个不同的值的$r\in\{0,1,\dotsc,a-1\}$，$$\#\{n<X: c(n)\equiv r\pmod{a}\} > \frac{\log_2\log_3 X}{a}.$$我们得到类似的结果，凹组成的奇数长度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
