Combinatorial and arithmetic properties of nr-partitions: A unified analytic generalization

Abstract

This paper presents a comprehensive study of $n^r$-partitions, defined by the generating function $\prod_{n=1}^{\infty} (1 - q^n)^{-n^r}$, which interpolates analytically between ordinary partitions ($r=0$), plane partitions ($r=1$), and higher-dimensional combinatorial structures ($r\geq2$). Our main contributions include: (1) explicit formulas relating $p_{(r)}(n)$ to divisor sums $\sigma_{r+1}(n)$ and integer compositions; (2) novel recurrence relations and congruence properties modulo primes and composite numbers; (3) complete Bell polynomial representations; and (4) combinatorial interpretations through generating function techniques. For weighted restricted $n^r$-partitions into three parts, we establish exact quasi-polynomial formulas. The results extend classical partition theory while providing new connections between combinatorics and number theory. The methods developed here offer a unified framework for investigating partition-like structures across dimensions, with applications in enumerative combinatorics.