Note on partitions into polynomials with number of parts in an arithmetic progression

TL;DR

This paper investigates the distribution of partitions of integers into parts from polynomial-generated sequences and their relation to arithmetic progressions, revealing equidistribution properties under specific conditions.

Contribution

It establishes new equidistribution results connecting partitions with polynomial parts to those with parts in arithmetic progressions, under certain divisibility conditions.

Findings

Proves equidistribution between polynomial-based partitions and arithmetic progression partitions.

Identifies conditions on the polynomial ensuring the distribution properties.

Provides theoretical framework for understanding partitions with polynomial and arithmetic progression parts.

Abstract

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results between the number of partitions of an integer $n$ whose parts are taken from the sequence $\{f(\ell)\}_{\ell=1}^{\infty}$ and the number of parts of those partitions which are in a certain arithmetic progression.