A note on the restricted partition function $p_\mathcal{A}(n,k)$

TL;DR

This paper studies the arithmetic properties, periodicity, and distribution of the restricted partition function $p_ ext{A}(n,k)$ modulo an integer, with applications to special sequences and restricted $m$-ary partitions.

Contribution

It investigates the modular behavior and density bounds of the restricted partition function for arbitrary sequences and fixed parameters, extending understanding of partition congruences.

Findings

Analyzed periodicity of $p_ ext{A}(n,k) mod m$ sequences.

Established bounds for the density of residue classes.

Applied results to specific sequences and restricted $m$-ary partitions.

Abstract

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set $\{n\in\mathbb{N}: p_\mathcal{A}(n,k) \equiv i \pmod{m}\}$ for a fixed positive integer $k$ and $i\in\{0,1,\ldots, m-1\}$. In particular, we apply our results to the special cases of the sequence $\mathcal{A}$. Furthermore, we present some results related to restricted $m$-ary partitions.