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The asymptotic behavior of the rectangle partition function $p(m,n)$
T0 review · reviewed 2026-08-28 · deepseek-v4-flash
Pith's one-line read For every fixed width m, the rectangle partition number p(m,n) satisfies log p(m,n) = π√(2mH_m/3)√n + O(log n), settling the conjecture.
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
The main result gives the growth rate of p(m,n) when the width m is fixed and the height n grows. The logarithm of p(m,n) is about a constant times the square root of n, plus a correction of size log n. The constant is expressed through the harmonic numbers H_m. The proof has two parts. For the upper bound, the authors count all multisets of rectangular blocks whose total area is mn, without checking whether they fit together. The generating function for these multisets factors into products of simple geometric series, and a short estimate gives the right exponential rate. For the lower bound, they construct a large family of genuine tilings. They reserve vertical columns for blocks of each possible smaller side, pack the blocks into horizontal strips inside the columns, and fill all leftover area with unit squares.
The construction is injective: different choices of the input partitions produce different multisets of blocks, so the number of tilings is at least the product of the sizes of the input families. An inequality involving the harmonic numbers guarantees that the columns fit side by side. The result confirms a conjecture from the authors' earlier work and extends earlier proofs for widths 2 and 3 to every fixed width.
Extended reading notes
Core claim
Theorem 1.1 states that for every fixed positive integer m, as n→∞, log p(m,n)=π√(2mH_m/3)√n+O(log n), and moreover p(m,n)≤exp(π√(2mH_m/3)√n) for all n≥m. If the paper is correct, the conjecture from [2] is true in the stronger logarithmic-error form, and the leading constant is exactly π√(2mH_m/3).
Load-bearing premise
The lower bound in Section 4 rests on Lemma 3.4, which asserts that the family A(A,l) of partitions with parts restricted to {A,...,ν_l} and total size at most σ_l still has size exp(π√(2/3)√l+O(log l)). This lemma imports Hardy-Ramanujan growth into the restricted family; if the restriction cost more than a power of l, the product construction would yield too few tilings and the main constant would not match. The lemma is proved, but it is the most fragile load-bearing step in the paper.
Formalized claims in Lean
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Claim #1: Theorem 1.1 states that for every fixed positive integer m, as n→∞, log p(m,n)=π√(2mH_m/3)√n+O(log n), and moreover p(m,n)≤exp(π√(2mH_m/3)√n) for all n≥m. If the paper is correct, the conjecture from [2] is true in the stronger logarithmic-error form, and the leading constant is exactly π√(2mH_m/3).
/-- @claim 1 Theorem 1.1 states that for every fixed positive integer m, as n→∞, log p(m,n)=π√(2mH_m/3)√n+O(log n), and moreover p(m,n)≤exp(π√(2mH_m/3)√n) for all n≥m. If the paper is correct, the conjecture from [2] is true in the stronger logarithmic-error form, and the leading constant is exactly π√(2mH_m/3). -/ def central_claim : Prop :=
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Editorial analysis
A structured set of objections, weighed in public.
Assumptions & free parameters
assumptions (3)
- standard math Hardy-Ramanujan asymptotic formula p(t) = (1/(4√3 t)) exp(π√(2t/3))(1+O(t^{-1/2}))
- standard math Euler's identity ∑_{k=1}∞ 1/k^2 = π^2/6
- standard math Multiset generating functions factor into Euler products: ∑_{ℓ≥0} U_m(ℓ)x^ℓ = ∏_{a=1}^m ∏_{b=a}∞ (1-x^{ab})^{-1}
Cite this review
Pith. "Pith review of The asymptotic behavior of the rectangle partition function $p(m,n)$." pith.science (2026). https://pith.science/paper/5UYBJUCK
@misc{pith2026260822955,
author = {Pith},
title = {Pith review of: The asymptotic behavior of the rectangle partition function $p(m,n)$},
year = {2026},
howpublished = {\url{https://pith.science/paper/5UYBJUCK}},
note = {Machine review of arXiv:2608.22955}
}
abstract
Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=\pi\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy--Ramanujan formula for integer partitions.
Figures
Reference graph
Works this paper leans on
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[1]
K. Gajdzica, R. Visser and M. Zakarczemny,Rectangle partitions generalizing integer partitions, Annals of Combinatorics (2026)
work page 2026
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[2]
A note on the partition function of a rectangle
K. Gajdzica and M. Zakarczemny,A note on the partition function of a rectangle,https://arxiv. org/abs/2608.08762v1[math.CO], preprint
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[3]
G. H. Hardy and S. Ramanujan,Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2)17(1918), 75–115. Theoretical Computer Science Department, F aculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland Email address:krystian.gajdzica@uj.edu.pl Department of Applied Mathematics, F aculty of...
work page 1918
Reviewed August 28, 2026 · model on record in the stance chip above.
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