Rectangle partitions generalizing integer partitions

TL;DR

The paper extends classical partition theory to rectangle tilings by introducing the rectangle partition function $p(m,n)$, notably tackling $p(2,n)$. It derives a Hardy–Ramanujan–type asymptotic $p(2,n)\sim \frac{\pi 2^{1/4}}{32}\,n^{-7/4}\exp(\pi\sqrt{2n})$ and shows Benford behavior for this sequence in all bases. It then studies horizontally symmetric tilings, restricted partitions $p_{k,l}(2,n)$ with exact small cases and a natural recurrence, and conjectures quasi-polynomial forms for $p_{k,1}(2,n)$, extending the classical theory to higher dimensions. Finally, it develops a rectangular generalization of $m$-ary partitions $b_{i,j}(2,n)$ and proves Alkauskas-type congruences modulo $m$, including explicit base-$m$ digit formulas. Together, these results weave generating-function, asymptotic, and modular perspectives into a cohesive higher-dimensional analogue of integer partitions.

Abstract

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a rectangle are indistinguishable if they consist of the same multiset of blocks, their geometric arrangement does not matter.

Figures (1)

• Figure 1: A partition of the rectangle $2\times 12$ into $4$ blocks $1\times3$ and $3$ blocks $1\times4$.

Theorems & Definitions (52)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Remark 1.4
• Example 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• proof
• Corollary 2.2