Self-avoiding polygons on a three-row square-lattice strip

Michael von Thaden

REVIEW 2 major objections 4 minor 8 references

Self-avoiding polygons on a three-row square-lattice strip

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The number of self-avoiding polygons on the three-row square-lattice strip is given by a closed formula built from an exponential term and a 4-periodic correction, and the type-2 subtype also equals a binomial sum, yielding a geometric proo

desk verdict Clean elementary enumeration of a known result; the delicate kink bijection is correct but under-proven. read the letter →

arxiv 2607.27397 v1 pith:WLZH5NPU submitted 2026-07-29 math.CO math-phmath.MP

Michael von Thaden This is my paper · ORCID

classification math.COmath-phmath.MP MSC 05A1505A19

keywords self-avoidingpolygonssquarelatticethree-rowstripclosedformulabinomialidentityA007909kinkdecompositionboundaryconfigurations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives a closed formula for the number of self-avoiding polygons on the three-row square-lattice strip, counting polygons up to horizontal translation and with reflections counted separately. The main result expresses the total count as a simple exponential in n/2 plus a 4-periodic correction term; the same structure appears for the four subtypes distinguished by how many vertical steps sit at the leftmost and rightmost columns. A second enumeration of the type-2/type-2 subtype as a binomial sum proves an identity previously evaluated in the literature, and the subtype matches the integer sequence A007909. The derivation is elementary and geometric, avoiding generating functions.

What carries the argument

The load-bearing structure is the unit-width slice: each slice between consecutive columns contains exactly two horizontal edges of the polygon. Around this, the paper builds two classifications: boundary type (one or two vertical steps at each extreme column) gives the recurrences, and the 'kink' decomposition — an interior kink moves the two horizontal edges from the extreme rows to an adjacent row-pair and back — gives the binomial sum and the binary-block encoding into compositions.

What would settle it

Brute-force enumerate all self-avoiding polygons on the three-row strip up to length 30 and compare p_{22}(2m+6) against (2^{m+1}+ρ(m))/5 and against the binomial sum; any mismatch for m=5 or m=6 would refute. More directly, search for a type-2/type-2 polygon in which two kinks share an interior x-coordinate or in which a kink starts from the inner row-pair rather than from the extreme rows, since such a polygon would invalidate the independent-kink decomposition.

Watch

Extended reading notes

Core claim

The paper establishes that on the strip S2, every self-avoiding polygon has exactly two horizontal edges in each vertical slice, forcing a 'key-bit' shape. Classifying polygons by whether the leftmost and rightmost columns contain one or two vertical steps yields coupled recurrences whose explicit solutions are (2^{n/2} terms plus a 4-periodic ρ) divided by 5. For the subtype with two vertical steps at both ends, the paper derives a binomial-sum formula by decomposing the polygon into kinks that switch the active pair of rows; equating the two formulas gives a geometric proof of a known binomial identity, and the counts coincide with the integer sequence A007909.

Load-bearing premise

The type-2 theorem assumes that every polygon with two vertical steps at both ends can be cut, left to right, into independent kinks that never interact or produce self-intersections, so that counting reduces to freely choosing 2i interior columns and i row choices; this exhaustiveness and independence is asserted without proof.

Editorial extensions

If this is right

For every even n, the total number of polygons on the strip can be read off directly from n mod 4 and n/2, with no recursion or generating functions.
The type-2/type-2 polygons of length 2m+6 are equinumerous with compositions of m into parts 1 of one kind and parts at least 3 of two kinds, giving a geometric interpretation of that combinatorial family.
Equating the closed form and the binomial sum yields a new geometric proof of the unweighted binomial identity previously evaluated in the literature.
The closed formulas imply that the connective constant for self-avoiding polygons on the three-row strip is sqrt(2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The block-decomposition into admissible binary words suggests a direct uniform sampling procedure for type-2 polygons: pick a composition of m, then random bits for each block; this is a concrete Monte Carlo tool the paper does not spell out.
The same slice-and-kink machinery might extend to strips of width 3 or 4, where the 4-periodic correction would presumably be replaced by a longer periodic correction; this is an untested extrapolation, not a paper claim.
Equating the two enumeration methods gives a bijective route to the unweighted binomial identity, so a weighted version of the same block encoding could recover the original weighted result treated in the literature, a step the paper only gestures at.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Quick take: this is a solid, honest piece of enumerative combinatorics. The counts themselves were already in Bousquet-Mélou and Brak's generating functions, and the paper says so. What's new is the package: a generating-function-free derivation, closed formulas for boundary subtypes, a binomial sum, and a bijective encoding that gives a geometric meaning to A007909 and a new proof of Larsen's identity. That's a real contribution, though narrow.

The recurrences in Lemma 2.1 are plausible and the algebra in Theorem 2.1 and Corollary 2.1 checks out. The formulas reproduce the initial values. The connection to A007909 is neat. The binary encoding in Section 2 is the strongest part: it gives a clean block decomposition and proves the composition interpretation directly. I agree with the reader: the delicate point is Theorem 2.2's kink decomposition. The proof is a paragraph and refers to a figure, but the stress-test note is right — the decomposition is sound. If you model each slice by the pair of rows carrying its horizontal edges, then the possible transitions are {0,2} -> {0,1} or {1,2}, and back. A direct {0,1}->{1,2} switch would force a step from row 0 to row 2, which doesn't exist. So the kinks are independent and the binomial count is correct. The converse — every choice of coordinates and row choices gives a single SAP — also holds, because tracing from the bottom left boundary vertex visits every edge. That argument should be in the paper as a lemma, not left to the reader.

Other soft spots are minor. Lemma 2.1's slice-peeling is also geometric and would benefit from a formal statement. The initial values in Corollary 2.1 are asserted; they're small and correct, but a table would help. And the remark that analogous formulas hold for other boundary types is unproved, though it follows from the same machinery.

Who's this for? People who work on exactly solved lattice models and like bijective proofs. It's not going to change statistical physics, but it's a clean classroom-level result and a nice example of a geometric proof replacing generating functions. I'd send it to a serious referee, mainly so they can ask for a formal lemma in Theorem 2.2 and a bit more detail in Lemma 2.1. With those revisions it's publishable.

Referee Report

2 major / 4 minor

Summary. The paper enumerates self-avoiding polygons (SAPs) on the three-row square-lattice strip S_2 = Z x {0,1,2}, counting edge sets up to horizontal translation and reflections separately. It introduces type-1 and type-2 boundary configurations determined by the number of vertical steps in the leftmost and rightmost columns, derives a two-recurrence system (Lemma 2.1), obtains closed forms for the subtype counts and for the total count as (2^{n/2}+periodic term)/5 (Corollary 2.1), and proves a binomial-sum formula for the type-2-to-type-2 count p^{S_2}_{22}(n) (Theorem 2.2). Finally, a binary encoding shows that p^{S_2}_{22}(2m+6) equals the OEIS sequence A007909, and hence equals the number of compositions of m into parts of size 1 of one kind and parts of size at least 3 of two kinds. The derivation is purely combinatorial and avoids generating functions, and comparing the closed form with the binomial sum gives a geometric proof of a binomial identity due to Larsen.

Significance. If the geometric arguments are made fully rigorous, the paper provides a clean, elementary enumeration of a nontrivial strip model. The recurrences and initial values are internally consistent, and the closed formulas check against small values n=4,...,12. The paper also gives an independent bijective proof of a binomial identity and a new combinatorial interpretation of A007909. These are concrete, falsifiable results. The main weaknesses are not circularity or algebraic errors but the informality of two geometric bijections, especially the kink decomposition in Theorem 2.2; the stress-test reasoning indicates the assertions are correct but they need to be written as formal lemmas for the paper to be fully convincing.

major comments (2)

[Theorem 2.2, proof (and Definition 2.2)] The proof of the binomial-sum formula rests on the assertion that every type-2-to-type-2 SAP decomposes into i independent 'complete kinks', and that every choice of 2i interior x-coordinates together with a row choice for each kink gives exactly one SAP. This is the load-bearing bijection of the theorem, but it is not formally proved. In particular, the text does not rule out a direct switch between the two adjacent-row configurations (rows {0,1} and {1,2}) in consecutive slices, nor does it prove that the graph assembled from an arbitrary choice of coordinates and row choices is a single self-avoiding cycle. A state-transition lemma describing the possible row-pair changes between consecutive unit-width slices would settle both points; the authors should include such a lemma or an equivalent formal argument.
[Lemma 2.1, proof] The recurrence system for the subtype counts is derived by 'peeling off' the rightmost unit-width slice and closing the resulting open path. The two cases (extreme-row pair vs. adjacent-row pair) are described intuitively and with reference to figures, but the claimed bijections and their inverses are not formalized. Since Corollary 2.1 and Theorem 2.1 depend entirely on these recurrences, the proof needs a precise statement of the maps between SAPs of length n and SAPs of length n-2 or n-4, including a demonstration that the inverse extension always yields a valid SAP. Without this, the recurrences are only verified by pictures and examples.

minor comments (4)

[Corollary 2.1, equations (3)-(6)] The formulas use the notation '2n/2' which is typographically ambiguous; these should be typeset as 2^{n/2}, 2^{n/2-1}, and 2^{n/2-2} respectively to avoid confusion.
[Section 2, figures] The proof frequently refers to Figures 2.1-2.4 for the key geometric operations. In the submitted text these figures were not available. The authors should ensure the figures are present and, more importantly, that the accompanying formal arguments do not rely on unreadable details of the figures.
[Section 2, after Theorem 2.2] The sentence 'Analogous statements also hold true for all SAPs with arbitrary left and right boundary configurations as well as for the overall number' is vague and not proved. Either provide the analogous statements explicitly or state clearly that they are not needed and are omitted.
[Definition 2.2] The term 'complete kink' is used heavily in the proof of Theorem 2.2 but is not formally defined. A precise definition (e.g., in terms of the row-pair of horizontal steps in consecutive slices) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the enumerations are derived from geometric recurrences and independent bijections; OEIS/Larsen identities are external comparisons, not inputs.

full rationale

The derivation chain is self-contained. Lemma 2.1 establishes the recurrences by explicit geometric peel-off bijections, and Theorem 2.1 is only algebraic manipulation of those recurrences. Corollary 2.1 proves the closed formulas by checking initial values and verifying the recurrences; this is a standard induction using uniqueness of solutions to linear recurrences, not an assumption of the formulas. Theorem 2.2 furnishes an independent enumeration of p_{22}(n) via a kink-coordinate bijection; the binomial sum is derived from that bijection, and only afterward is it compared with A007909/Larsen. Although the kink-decomposition lemma is stated informally and would benefit from a formal proof, that is a rigor gap, not circularity: no target quantity is used to define or fit the kink parameters, and no self-citation carries the argument. The references to Bousquet-Mélou–Brak are contextual, and the OEIS/Larsen identities are external checks rather than premises.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical fit: the 4-periodic function ρ is forced by the four initial values and the recurrences, not fitted to data. No new physical or combinatorial entities are introduced. The listed axioms are the unproved geometric assertions the derivation rests on.

assumptions (4)

domain assumption Every unit-width slice between consecutive columns inside the horizontal span of an SAP on S2 contains exactly two horizontal edges.
Remark 2.1 states this as an observation; it underpins the slice-peeling recurrences and the 2l horizontal-step count in Theorem 2.2.
domain assumption The leftmost/rightmost column configurations of every SAP on S2 are exhausted by type-1 (one vertical step) and type-2 (two vertical steps).
Definition 2.2 assumes this exhaustiveness without proof; the whole recurrence classification depends on it.
domain assumption The peeling-and-closing operations in Lemma 2.1 are bijections with unique inverse extensions.
The lemma's proof asserts bijectivity based on figures; no formal injection/surjection analysis is given.
domain assumption Every type-2 SAP decomposes into i independent, non-overlapping kinks, each a type-2 → type-1 → type-2 excursion, with independent row choices.
Theorem 2.2 proof depends on this decomposition to produce the binomial sum; exhaustiveness and non-nesting are not formally proven.

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Pith. "Pith review of Self-avoiding polygons on a three-row square-lattice strip." pith.science (2026). https://pith.science/paper/WLZH5NPU

@misc{pith2026260727397,
  author       = {Pith},
  title        = {Pith review of: Self-avoiding polygons on a three-row square-lattice strip},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLZH5NPU}},
  note         = {Machine review of arXiv:2607.27397}
}

abstract

We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\mathbb{Z}\times\{0,1,2\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new geometric proof of an identity arising in Larsen's treatment \cite{L07} of a problem posed by Gessel \cite{G95}. Finally, we show that this subtype of SAPs is closely connected to the sequence A007909. More precisely, for $m\geq0$, the number of these SAPs whose leftmost and rightmost columns each contain two vertical steps and whose length equals $2m+6$ is given by the term of this sequence with index $m$, which thereby acquires a geometric interpretation alongside the compositions it enumerates.

Figures

Figures reproduced from arXiv: 2607.27397 by the authors.

**Figure 2.1.** An SAP resembling a key bit. type-1 configuration type-2 configuration [PITH_FULL_IMAGE:figures/full_fig_p002_2_1.png] view at source ↗

**Figure 2.2.** SAPs with type-1 and type-2 configurations, respectively, at both extreme x-coordinates. Furthermore, we observe that the length n of every SAP is always an even number, because an SAP always returns to its origin, which means there must be the same number of vertical steps going “up” as steps going “down”. The same holds true for the number of horizontal steps going to the “right” and to the “left”. For our approac… view at source ↗

**Figure 2.3.** Deletion of the two possible right-hand configurations and closure of the resulting open polygonal paths. It is important to point out that these recursions are equivalent to the transfer structure implicit in the generating function of Bousquet-M´elou and Brak [2], from which the closed formula below could also be extracted by routine computations; our derivation is instead purely geometric and combinatorial. Base… view at source ↗

Figures from the paper (1 more)

**Figure 2.4.** Figure 2.4: A self-avoiding polygon of length 24 with type-2 boundary configurations at both extreme x-coordinates. Its binary code is 001111001, with the unique block decomposition 0|0|111|1001. The blocks 111 and 1001 encode the upper and lower kink, respectively. There is the…

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Reference graph

Works this paper leans on

8 extracted references

[1]

A. T. Benjamin, Self-avoiding walks and Fibonacci numbers,Fibonacci Quart.44(2006) 330–334

2006
[2]

Bousquet-M´ elou, R

M. Bousquet-M´ elou, R. Brak, Exactly solved models of polyominoes and polygons, In: A. J. Guttmann (Ed.),Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, vol. 775, Springer Dordrecht, 2009, 43–78

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[3]

I. M. Gessel, Problem 10424,Amer. Math. Monthly102(1995) 70

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[4]

A. J. Guttmann, I. Jensen, Effect of confinement: polygons in strips, slabs and rectangles, In: A. J. Guttmann (Ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, vol. 775, Springer Dordrecht, 2009, 235–246

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[5]

M. E. Larsen,Summa Summarum, A K Peters, Wellesley, MA, 2007

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[6]

Madras, G

N. Madras, G. Slade,The Self-Avoiding Walk, Modern Birkh¨ auser Classics, Birkh¨ auser/Springer, New York, 2013, reprint of the 1993 edition

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[7]

OEIS Foundation Inc.,The On-Line Encyclopedia of Integer Sequences, Sequence A007909,https://oeis.org/ A007909, accessed 29 July 2026

2026
[8]

Zeilberger, Self-avoiding walks, the language of science, and Fibonacci numbers,J

D. Zeilberger, Self-avoiding walks, the language of science, and Fibonacci numbers,J. Statist. Plann. Inference54 (1996), 135–138. 6

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

A. T. Benjamin, Self-avoiding walks and Fibonacci numbers,Fibonacci Quart.44(2006) 330–334

2006

[2] [2]

Bousquet-M´ elou, R

M. Bousquet-M´ elou, R. Brak, Exactly solved models of polyominoes and polygons, In: A. J. Guttmann (Ed.),Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, vol. 775, Springer Dordrecht, 2009, 43–78

2009

[3] [3]

I. M. Gessel, Problem 10424,Amer. Math. Monthly102(1995) 70

1995

[4] [4]

A. J. Guttmann, I. Jensen, Effect of confinement: polygons in strips, slabs and rectangles, In: A. J. Guttmann (Ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, vol. 775, Springer Dordrecht, 2009, 235–246

2009

[5] [5]

M. E. Larsen,Summa Summarum, A K Peters, Wellesley, MA, 2007

2007

[6] [6]

Madras, G

N. Madras, G. Slade,The Self-Avoiding Walk, Modern Birkh¨ auser Classics, Birkh¨ auser/Springer, New York, 2013, reprint of the 1993 edition

2013

[7] [7]

OEIS Foundation Inc.,The On-Line Encyclopedia of Integer Sequences, Sequence A007909,https://oeis.org/ A007909, accessed 29 July 2026

2026

[8] [8]

Zeilberger, Self-avoiding walks, the language of science, and Fibonacci numbers,J

D. Zeilberger, Self-avoiding walks, the language of science, and Fibonacci numbers,J. Statist. Plann. Inference54 (1996), 135–138. 6

1996