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arxiv: 2604.12392 · v1 · submitted 2026-04-14 · 🧮 math.CO

Enumerations and Bijections for Stanley Polyominoes

Jean-Luc Baril , Aubrey Blecher , Jos\'e Luis ram\'irez This is my paper

Pith reviewed 2026-05-10 15:02 UTC · model grok-4.3

classification 🧮 math.CO
keywords Stanley polyominoesparallelogram polyominoesgenerating functionsbijectionsDyck pathsMotzkin pathscoin fountainsskew Ferrer diagrams
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The pith

Stanley polyominoes have generating functions for area and semiperimeter plus explicit bijections to Dyck paths, skew diagrams, and peakless Motzkin paths.

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

Stanley polyominoes form a subclass of parallelogram polyominoes whose rows each begin and end strictly farther right than the row above. The paper constructs generating functions that count them by columns and rows, by area, by semiperimeter, and by interior points and edges. It supplies direct bijections from these polyominoes to Dyck paths, skew Ferrer diagrams, and peakless Motzkin paths. One consequence is an explicit bijection between all parallelogram polyominoes of area n and coin fountains having n coins in every even row and n-k coins in the odd rows.

Core claim

Stanley polyominoes, defined by the strict rightward shift condition on successive rows, are placed in bijection with Dyck paths, skew Ferrer diagrams, and peakless Motzkin paths. From these correspondences the authors obtain ordinary generating functions that track the number of columns and rows, the area, the semiperimeter, and the counts of interior points and edges. The same machinery yields a bijection between parallelogram polyominoes of area n and coin fountains with n coins on even-numbered rows and n-k coins on odd-numbered rows, thereby settling a previously open question.

What carries the argument

The explicit bijections that map the row-strict geometry of Stanley polyominoes onto Dyck paths, skew Ferrer diagrams, and peakless Motzkin paths, together with the algebraic extraction of the corresponding generating functions.

If this is right

  • The number of Stanley polyominoes of any given area or semiperimeter is given by the appropriate coefficient in the derived generating functions.
  • Any known enumeration or statistic on Dyck paths or peakless Motzkin paths transfers directly to a corresponding count or statistic on Stanley polyominoes.
  • Parallelogram polyominoes of area n stand in explicit bijection with the specified family of coin fountains.
  • Statistics such as the number of interior points or edges admit combinatorial interpretations via the path or diagram side of each bijection.

Where Pith is reading between the lines

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

  • The same row-strict restriction may admit similar bijections for other natural statistics or for related families such as directed column-convex polyominoes.
  • The coin-fountain bijection suggests that geometric properties of parallelogram polyominoes can be reinterpreted as coin-arrangement constraints without reference to the polyominoes themselves.
  • The generating functions may be refined further by adding variables for additional geometric features that are preserved by the bijections.

Load-bearing premise

The bijections and generating-function derivations are constructed without error from the row-strict definition and the standard definitions of the target objects.

What would settle it

A concrete mismatch, for any fixed small n, between the coefficient of x^n in one of the derived generating functions and the number of Stanley polyominoes of area n (or semiperimeter n) obtained by exhaustive enumeration would falsify the claims.

Figures

Figures reproduced from arXiv: 2604.12392 by Aubrey Blecher, Jean-Luc Baril, Jos\'e Luis ram\'irez.

Figure 1
Figure 1. Figure 1: A Stanley Polyomino and its associated Dyck path by ϕ. We now introduce several classical statistics on polyominoes. We refer to [19] for a histor￾ical review of polyominoes, and to [18] for definitions of several statistics and enumerative methods related to polyominoes. Let P be a Stanley polyomino. We denote by first(P) the number of cells in the first row. We denote by col(P) and row(P) the number of c… view at source ↗
Figure 2
Figure 2. Figure 2: Decomposition according to the cell numbers of the first two rows. After applying operation (1), the new bottom row contributes one new row, i new cells, i − 1 new interior points, and i − 2 new strictly internal edges when i ≥ 2. Hence the corresponding weight is α1 = yzu and αi = yzip i−2 q i−1u i with i ≥ 2. Therefore, summing over i = 1, . . . , h − 1 and then over h, we obtain A2 := X h≥2 Fhα1 + X h≥3… view at source ↗
Figure 3
Figure 3. Figure 3: The five Stanley polyominoes with four columns. From left to right, the contributions are respectively yz4x 4 , y 2 z 5x 4 , qy2 z 6x 4 , y 2 z 5x 4 , and y 3 z 6x 4 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The bijection χ maps each of the four peakless Motzkin paths with four steps to the corresponding Stanley polyomino of semiperimeter 6. The number of steps ending on the x-axis corresponds to the number of cells in the first row minus one. Now, substituting the above expression for G(1) into (3) and cancelling the factor (u−r), we obtain the following result. Theorem 2.6. The generating function for Stanle… view at source ↗
Figure 5
Figure 5. Figure 5: The eight Stanley polyominoes with semiperimeter 7. Their contributions to G(u) are, from left to right, u 6x 7 , u 4x 7 , u 4x 7 , u 4x 7 , u 3x 7 , u 3x 7 , u 2x 7 , and u 2x 7 . The corresponding preimages under χ ′ are shown below, in the same order. By calculating ∂u(G(u))|u=1 we obtain the following [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The six Stanley polyominoes with area 6. We now define recursively a map f from C to S that sends Cn into Sn+1, and we will prove that f sends the statistic 2e(C) − o(C) into the area. Definition 2.10. If C consists of a single coin (that is, if n = 1), then f(C) is the unique Stanley polyomino in S2 consisting of a single row with two cells. Now let C be a coin fountain with at least two coins, and let di… view at source ↗
Figure 7
Figure 7. Figure 7: The two cases in the recursive construction of f. The previous lemma shows that f is a well-defined map from C to S, and that f(C) has d + 1 columns whenever C is a coin fountain with d diagonals. A simple induction shows that f is injective: indeed, there do not exist integers ℓ1 and ℓ2 such that both firstD(P) = ℓ1, firstR(P) ≥ ℓ1 + 2, and firstD(P) ≥ ℓ2 + 1, firstR(P) = ℓ2 + 2 hold simultaneously. Since… view at source ↗
Figure 8
Figure 8. Figure 8: A coin fountain C such that 2e(C) − o(C) = 21, together with its image under f. As a byproduct, this bijection allows us to address an open problem posed by Bala (see [1]), namely the construction of an explicit bijection between coin fountains and skew Ferrers diagrams [12] (equivalently, parallelogram polyominoes). Let Pn be the set of parallelogram polyominoes of area n and let Cn be the set of coin fou… view at source ↗
Figure 9
Figure 9. Figure 9: A parallelogram polyomino of area 24 and the construction of its image by ψ. 3. Continued fraction point of view The expression obtained in Theorem 2.9 can also be written as a continued fraction by means of the bijection ϕ between Stanley polyominoes with n+ 1 columns and Dyck paths of semilength n (see the introduction). For background on the connection between lattice paths and continued fractions, see … view at source ↗
Figure 10
Figure 10. Figure 10: The nine Dyck paths with sum of peak heights equal to 4. Their contributions to A(p, q, v) are, from left to right, p 4 , p 3 , p 3 , p 3 , p 2 , p 2 , p 2 , p 2 v, and p. Setting p = q and v = 1 in A(p, q, v), we obtain A(q, q, 1) = F(1, 1, q, 1, 1). Hence the following corollary gives an alternative expression for F(1, 1, q, 1, 1) as a continued fraction. Corollary 3.2. The generating function F(1, 1, q… view at source ↗
Figure 11
Figure 11. Figure 11: The five Dyck paths with sum of peak heights plus number of peaks equal to 6. Corollary 3.4. Let A(1, q, 1) be the generating function for Dyck paths, where q marks the sum of the peak heights. Equivalently, A(1, q, 1) is the generating function for Stanley polyominoes with respect to the area minus the number of rows. Then A(1, q, 1) = −1 + 1 2 − q − 1 2 − q 2 − 1 2 − q 3 − 1 2 − q 4 − · · · . The first … view at source ↗
read the original abstract

Stanley polyominoes are a subclass of parallelogram polyominoes in which each row begins strictly to the right of the beginning of the previous row and ends strictly to the right of the end of the previous row. In this paper, we derive generating functions for Stanley polyominoes based on the numbers of columns and rows, area, semiperimeter, and numbers of interior points and edges. We also establish combinatorial connections through bijections with other combinatorial structures such as Dyck paths, skew Ferrer diagrams, and peakless Motzkin paths. As a byproduct, we answer the open question of finding a bijection between parallelogram polyominoes of area $n$ and coin fountains with $n$ coins in the even-numbered rows and $n-k$ coins in the odd-numbered rows.

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.

Referee Report

0 major / 2 minor

Summary. The paper defines Stanley polyominoes as a row-strict subclass of parallelogram polyominoes (each row begins and ends strictly to the right of the preceding row) and derives generating functions for these objects tracked by the number of columns and rows, area, semiperimeter, interior points, and edges. It constructs explicit bijections with Dyck paths, skew Ferrers diagrams, and peakless Motzkin paths, and as a byproduct supplies a bijection between parallelogram polyominoes of area n and coin fountains having n coins in even-numbered rows and n-k coins in odd-numbered rows.

Significance. If the recursive decompositions, generating-function derivations, and bijective constructions hold, the work supplies new enumerative tools for a natural subclass of polyominoes and resolves an open combinatorial question. The explicit, invertible mappings to classical path and diagram objects allow transfer of known results and statistics, strengthening the combinatorial understanding of these structures.

minor comments (2)
  1. [Section 3] The initial conditions and base cases for the recursive decompositions used to obtain the generating functions are stated only implicitly; an explicit listing (e.g., for the empty or single-row cases) would improve readability.
  2. [Section 5] Figure 4, which illustrates the composite bijection to coin fountains, would benefit from an additional panel or caption step that verifies preservation of the area statistic.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on Stanley polyominoes and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are explicit and self-contained

full rationale

The paper constructs generating functions and bijections directly from the given row-strict definition of Stanley polyominoes via recursive decompositions and invertible mappings to Dyck paths, skew Ferrers diagrams, and peakless Motzkin paths. These steps preserve the tracked statistics (area, semiperimeter, interior points/edges) by explicit construction and verification of bijectivity, without reducing to fitted parameters, self-referential equations, or load-bearing self-citations. The composite bijection for the coin-fountain question is built from the established maps and resolves the open problem independently. No step equates a claimed result to its inputs by definition or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5441 in / 1225 out tokens · 72603 ms · 2026-05-10T15:02:31.908392+00:00 · methodology

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Reference graph

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