Perfect Overpartitions and Factorization of Integers

TL;DR

This paper extends MacMahon's notion of perfect partitions to perfect overpartitions, where the last occurrence of a part may be overlined. It establishes a bijection between perfect overpartitions of $n$ and ordered factorizations of $n+1$, with the presence of a factor $2$ corresponding to an overlined part, so overlines occur precisely when $n+1$ admits a factor $2$ and the weight $n$ can be odd. The authors derive a compact counting framework: $\overline{\mathrm{pp}}(n,r)=\sum_{v=r}^s \binom{v}{r} f_v(n+1)$ and $\overline{\mathrm{pp}}(n)=\sum_{r=0}^s\sum_{v=r}^s \binom{v}{r} f_v(n+1)$, where $f_v(N)$ counts ordered factorizations of $N$ with $v$ copies of $2$; they decompose $f_v(N)$ into three classes to enable recurrences and closed forms. The work provides explicit formulas for small $2$-adic order, examples, and algorithmic recurrences (computable in Maple), contributing exact combinatorial counts and sequences for perfect overpartitions and showing how parity and factorization structures govern the enumeration.

Abstract

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply the same definition to overpartitions which are integer partitions with the additional property that the final occurrence of each part may be overlined. It turns out that perfect overpartitions are enumerated by ordered factorization functions in which the occurrence of 2 as a factor determines the presence of an overlined part.

Theorems & Definitions (18)

• Theorem 1
• proof : Proof of Theorem \ref{['thm1']}
• Example
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• Theorem 2
• proof