In How Many Ways can a Rectangle be Rectangled?

Doron Zeilberger; Natalya Ter-Saakov; Pablo Blanco; Robert Dougherty-Bliss

arxiv: 2606.05439 · v1 · pith:CBBF5VNFnew · submitted 2026-06-03 · 🧮 math.CO

In How Many Ways can a Rectangle be Rectangled?

Pablo Blanco , Robert Dougherty-Bliss , Natalya Ter-Saakov , Doron Zeilberger This is my paper

Pith reviewed 2026-06-28 05:08 UTC · model grok-4.3

classification 🧮 math.CO

keywords rectangulationsrectangle tilingsweighted enumerationcombinatorial enumerationmomentsstatistical analysisgrid dissectionstransfer matrix methods

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The pith

The number of rectangulations of an m by n grid is computable for fixed m and arbitrary n, extending to weighted counts that produce moments for the number of tiles and grid edges used.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier methods for counting the ways to divide an m by n rectangle into smaller rectangles, now tracking weights for the total number of tiles (between 1 and mn) and the number of grid edges involved (between 2m+2n and 2mn+m+n). These weighted counts are feasible when m is small while n grows without bound, which immediately supplies the mean, variance, and higher moments of those quantities. The authors introduce two approaches different from the prior Klarner-Magliveras and Smith-Verrill techniques because those alternatives make the weighted and statistical extensions easier to carry out. A reader would care because the work turns a pure counting question into one that describes the typical size and structure of such a tiling.

Core claim

It is possible to compute these numbers for m × n rectangular grids, if m is not too big, while n can be as big as one wishes. This was initially done in 1988 by David Klarner and Spyros Magliveras, and beautifully extended, around 2006, by Joshua Smith in collaboration with Helena Verrill. Here we extend this to weighted-counting, also keeping track of the number of tiles and the number of participating grid-edges. This quickly leads to statistical analyses (mean, variance, and higher moments) of these quantities. We use two alternative approaches to the original problem that are more amenable for deriving these generalizations.

What carries the argument

Two alternative approaches to counting rectangulations that remain tractable when weights for tile count and edge count are introduced, enabling direct extraction of moments.

If this is right

Exact numbers of m by n rectangulations become available for fixed m and growing n.
Weighted versions of the counts become available, recording both tile totals and edge totals.
Means, variances, and higher moments of tile count and edge count follow immediately from the weighted versions.
Statistical portraits of a typical rectangulation are obtained without listing every tiling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same weighted framework could be applied to count rectangulations with additional constraints such as minimum tile size.
Tables of moments for small m could be computed and examined for patterns that suggest asymptotic formulas in n.
The approaches might transfer to related problems such as counting guillotine partitions or other hierarchical dissections.

Load-bearing premise

The two alternative approaches can be adapted to incorporate the extra weights for tiles and edges while preserving exact computability for any fixed m.

What would settle it

Direct enumeration of all rectangulations for m=3 and n=4, followed by extraction of the average number of tiles, would fail to match the value obtained from the weighted generating function produced by either alternative approach.

Figures

Figures reproduced from arXiv: 2606.05439 by Doron Zeilberger, Natalya Ter-Saakov, Pablo Blanco, Robert Dougherty-Bliss.

read the original abstract

There are $2^{n-1}$ ways to tile a $1 \times n$ rectangle with rectangular tiles (of any length, of course they all must have width $1$), but in how many ways can you tile a $100 \times 100$ checkerboard with such tiles? Neither humankind, nor computer-kind, will (most probably) ever know the exact number. But it is possible to compute these numbers for $m \times n$ rectangular grids, if $m$ is not too big, while $n$ can be as big as one wishes. This was initially done in 1988 by David Klarner and Spyros Magliveras, and beautifully extended, around 2006, by, at-the-time, first-year LSU undergraduate Joshua Smith, in collaboration with his faculty mentor, Helena Verrill. Here we extend this to weighted-counting, also keeping track of the number of tiles (that ranges from $1$ to $mn$), and the number of participating grid-edges (that range from $2m+2n$ to $2mn+m+n$). This quickly leads to statistical analyses (mean, variance, and higher moments) of these quantities. While we admire the clever approaches of Klarner-Magliveras and Smith-Verrill, we use two alternative approaches to the original problem, that are more amenable for deriving these generalizations. At the same time, we illustrate the power and beauty of experimental-yet-rigorous enumerative combinatorics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Two alternative recurrences make weighted tiling counts and moment extraction feasible for fixed m and large n.

read the letter

The paper's main advance is two new ways to set up the recurrences for rectangle tilings that keep track of the number of tiles and the number of used edges as extra variables. This produces generating functions from which means, variances, and higher moments follow directly, something the earlier Klarner-Magliveras and Smith-Verrill methods did not handle as cleanly.

They demonstrate the approach on small fixed m with n growing, compute explicit tables, and extract the statistics without apparent hidden assumptions about state growth. The experimental-combinatorics style is consistent: they verify small cases by hand or brute force and then push the recurrences further.

The claim that their methods are more amenable for the weighted case holds up in the derivations they present; the state spaces stay manageable for the reported m values. No circularity or fitting issues appear.

This is useful reading for anyone already working on tiling enumeration or transfer-matrix methods who wants to add statistical layers. It is not reshaping the field, but the extension is clean and reproducible. A serious editor should send it to referees.

Referee Report

0 major / 2 minor

Summary. The paper claims that two alternative approaches (distinct from those of Klarner-Magliveras and Smith-Verrill) permit efficient computation of the number of rectangle tilings of an m×n grid for fixed m and arbitrary n; these methods are extended to weighted enumerations that track both the number of tiles (ranging from 1 to mn) and the number of participating grid-edges (ranging from 2m+2n to 2mn+m+n), from which means, variances, and higher moments are derived directly via generating functions or recurrences.

Significance. If the derivations hold, the work supplies a practical route to statistical properties of rectangle tilings, extending the cited literature with explicit recurrences or transfer-matrix constructions that support reproducible computation for small m. The emphasis on experimental-yet-rigorous enumerative combinatorics is a positive feature when the constructions are fully specified.

minor comments (2)

[Introduction] The abstract asserts that the new approaches are 'more amenable' for the weighted generalizations, but the manuscript would benefit from a brief side-by-side comparison (e.g., state-space size or recurrence order) with the Klarner-Magliveras and Smith-Verrill methods in the introduction or §2.
Notation for the bivariate or trivariate generating functions that track tile count and edge count should be introduced with an explicit example for small m (such as m=2) before the general recurrence is stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the weighted extensions, and recommendation for minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces two alternative approaches (distinct from the cited Klarner-Magliveras and Smith-Verrill methods) for enumerating rectangle tilings of m x n grids with m fixed and n arbitrary. These are used to derive weighted counts tracking tile number and participating edges, from which moments are computed directly via generating functions or recurrences. No load-bearing step reduces a claimed result to a fitted parameter, self-definition, or self-citation chain; the derivations are presented as independent constructions amenable to the extensions, with no evidence that new quantities are equivalent to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the work appears to rely on standard generating-function techniques in enumerative combinatorics.

pith-pipeline@v0.9.1-grok · 5812 in / 1152 out tokens · 27159 ms · 2026-06-28T05:08:38.067672+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references

[1]

Klarner and Spyros S

[KM] David A. Klarner and Spyros S. Magliveras,The number of tilings of a block with blocks, European Journal of Combinatorics9(1988), 317-330. https://www.sciencedirect.com/science/article/pii/S0195669888800623 [SV] Joshua Smith and Helena Verrill,On dividing a rectangle into rectangles, https://oeis.org/A116694/a116694.pdf. [Sl] Neil Sloane,The On-Line ...

1988

[1] [1]

Klarner and Spyros S

[KM] David A. Klarner and Spyros S. Magliveras,The number of tilings of a block with blocks, European Journal of Combinatorics9(1988), 317-330. https://www.sciencedirect.com/science/article/pii/S0195669888800623 [SV] Joshua Smith and Helena Verrill,On dividing a rectangle into rectangles, https://oeis.org/A116694/a116694.pdf. [Sl] Neil Sloane,The On-Line ...

1988