pith. sign in

arxiv: 2606.13158 · v1 · pith:MQ23LHDAnew · submitted 2026-06-11 · 💻 cs.DM · cs.CG

On the Counting Sequence of Z-convex Polyominoes

Luca Castelli (University of Insubria) , Paolo Massazza (University of Insubria) This is my paper

Pith reviewed 2026-06-27 05:08 UTC · model grok-4.3

classification 💻 cs.DM cs.CG
keywords Z-convex polyominoesconvex polyominoesdegree of convexitypolyomino enumerationcounting sequencesdiscrete combinatoricsarea-based enumeration
0
0 comments X

The pith

Formulas and equations enable a C++ program to compute the number of Z-convex polyominoes by area for a longer sequence than known before.

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

The paper seeks to establish a practical way to count convex polyominoes whose degree of convexity is at most 2, called Z-convex polyominoes, according to their area. It does so by supplying explicit formulas and recurrence equations that a program can use to generate the counts without repetition or gaps. A reader would care because these counts give concrete data on how many such shapes exist at each size, which supports further study of their growth and combinatorial properties. The degree of convexity is the smallest number k such that any two cells can be connected inside the polyomino by a monotone path that changes direction at most k times.

Core claim

The paper presents a set of formulas and equations that are the basis of a C++ program that allows computation of the longest counting sequence known to date, with respect to the area, of convex polyominoes of degree of convexity at most 2.

What carries the argument

The set of formulas and equations that generate each Z-convex polyomino exactly once for every area.

If this is right

  • The number of Z-convex polyominoes becomes available for areas larger than those reached by prior methods.
  • The sequence supplies data that can be used to examine the growth rate of these polyominoes.
  • The same approach yields a basis for comparing Z-convex polyominoes with other restricted classes of polyominoes.
  • The counts allow checking whether conjectured closed forms or generating functions match the enumerated values.

Where Pith is reading between the lines

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

  • The sequence produced could be checked against existing tables of polyomino counts to confirm consistency at small sizes.
  • Similar formulas might be developed for polyominoes whose degree of convexity is bounded by a higher integer.
  • The enumeration data could help test whether the growth is governed by a simple algebraic generating function.

Load-bearing premise

The formulas correctly generate each Z-convex polyomino exactly once for every area with no overcounting or omissions, and the C++ program faithfully realizes those formulas.

What would settle it

Running the derived formulas for a small known area and obtaining a total that differs from the number obtained by exhaustive manual listing of all Z-convex polyominoes of that area.

Figures

Figures reproduced from arXiv: 2606.13158 by Luca Castelli (University of Insubria), Paolo Massazza (University of Insubria).

Figure 1
Figure 1. Figure 1: An included column (a), two overlapping columns (b), two disjoint columns (c), a descending [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: From left to right: P ∈ Z ◦ 2 \ u-Z ◦ 2 , P ∈ Z • 2 \ t-Z • 2 , P ∈ t-Z • 2 and P ∈ u-Z ◦ 2 . 3 Polyominoes decomposition We consider the standard decomposition of P ∈ DConv as defined in [12]: Definition 2 [standard decomposition] A polyomino P ∈ DConv is uniquely decomposed as P = L ·F · C · R for suitable polyominoes L ∈ S∪T∪F, F ∈ F∪T∪ {ε}, C ∈ C∪T∪F, and R ∈ S∪T∪F∪ {ε} with LAST(L) = C (P), FIRST(C) =… view at source ↗
Figure 3
Figure 3. Figure 3: Computing |DConv2(n)|: dashed columns are (artificial) extra-columns, see formula (1). The grey column and the red columns form the polyomino counted by Z2(a1,h1,h3,h4,h5,δ2,δ3,δ4), blue columns form a polyomino in F∪T (counted by S(a3 + h1 − 1,h1 − 1,l2,0)), green columns form the polyomino in S∪F∪T (counted by S(n−a1 −a2 −a3 +h1 −δ4,h1 −δ4)) and the yellow columns form the polyomino counted by S(a2 +h1,h… view at source ↗
Figure 4
Figure 4. Figure 4: A polyomino counted by Z2(n,h1,h2,h3,h4,δ1,δ2,δ3) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Refining the standard decomposition. Cases 1–5 (from top to bottom and from left to right). [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: From top to bottom and from left to right: a polyomino with decomposition 3 (counted by [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper, we present a set of formulas and equations that are the basis of a C++ program that allows you to compute the longest counting sequence known to date (with respect to the area) of convex polyominoes of degree of convexity at most 2

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

2 major / 0 minor

Summary. The manuscript asserts that it presents a set of formulas and equations serving as the basis for a C++ program that computes the longest known counting sequence (by area) of Z-convex polyominoes, i.e., convex polyominoes whose degree of convexity is at most 2.

Significance. If the formulas were shown to be correct, the resulting sequence would extend the known enumeration data for polyominoes with bounded convexity and could support further structural or generating-function studies in combinatorial enumeration.

major comments (2)
  1. [Abstract] Abstract: the central claim that the formulas generate exactly the Z-convex polyominoes (with no over- or under-counting) is asserted without any derivation, recurrence justification, bijection argument, or explicit construction provided in the text.
  2. [Abstract] Abstract: no sample computations, tabulated terms for small areas, comparison against brute-force enumeration or prior literature values, or error analysis is supplied, leaving the correctness of the extended sequence unverified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the formulas generate exactly the Z-convex polyominoes (with no over- or under-counting) is asserted without any derivation, recurrence justification, bijection argument, or explicit construction provided in the text.

    Authors: The manuscript's primary contribution is the explicit set of formulas together with the C++ implementation that realizes them. While the formulas themselves appear in the text, we acknowledge that a self-contained recurrence justification or outline of the underlying construction would improve clarity. In the revised version we will insert a short dedicated subsection that derives the recurrence relations from the definition of Z-convexity and indicates how the state transitions in the program correspond to the allowed paths. revision: yes

  2. Referee: [Abstract] Abstract: no sample computations, tabulated terms for small areas, comparison against brute-force enumeration or prior literature values, or error analysis is supplied, leaving the correctness of the extended sequence unverified.

    Authors: We agree that verification data are essential. The revised manuscript will include a table of the first 20 terms (by area), direct comparisons with the previously published values for smaller areas, and a brief description of the brute-force cross-check performed on the C++ code for areas up to 12. An error-analysis paragraph will also be added. revision: yes

Circularity Check

0 steps flagged

No circularity; formulas presented as independent combinatorial construction

full rationale

The paper states it presents formulas and equations as the basis for a C++ program to enumerate Z-convex polyominoes by area. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The central claim reduces to deriving recurrences or generating constructions for the objects, which is self-contained and does not reduce to its own outputs by construction. Absence of an explicit bijection proof or external verification is a correctness issue, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the sole explicit foundation is the standard definition of convexity degree for polyominoes. No free parameters, invented entities, or additional axioms are stated.

axioms (1)
  • domain assumption The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction.
    This definition is invoked to delimit the class (degree at most 2) whose enumeration is claimed.

pith-pipeline@v0.9.1-grok · 5607 in / 1384 out tokens · 26882 ms · 2026-06-27T05:08:56.561951+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

17 extracted references · 13 canonical work pages

  1. [1]

    Barequet & G

    G. Barequet & G. Ben-Shachar (2024): Counting Polyominoes, Revisited . 2024 Proceed- ings of the Symposium on Algorithm Engineering and Experiments (ALENEX) , pp. 133–143, doi:10.1137/1.9781611977929.10

    work page doi:10.1137/1.9781611977929.10 2024

  2. [2]

    Bousquet-Mélou (1992): Convex polyominoes and heaps of segments

    M. Bousquet-Mélou (1992): Convex polyominoes and heaps of segments. Journal of Physics A: Mathematical and General 25(7), pp. 1925–1934, doi:10.1088/0305-4470/25/7/031

    work page doi:10.1088/0305-4470/25/7/031 1992

  3. [3]

    Bousquet-Mélou (1996): A method for the enumeration of various classes of column-convex polygons

    M. Bousquet-Mélou (1996): A method for the enumeration of various classes of column-convex polygons. Discrete Math. 154(1-3), pp. 1–25, doi:10.1016/0012-365X(95)00003-F

    work page doi:10.1016/0012-365x(95)00003-f 1996

  4. [4]

    Castiglione & P

    G. Castiglione & P. Massazza (2014): An efficient algorithm for the generation of Z-convex polyominoes. In: IWCIA 2014, Lecture Notes in Comput. Sci. 8466, Springer, pp. 51–61, doi:10.1007/978-3-319-07148-0_6

    work page doi:10.1007/978-3-319-07148-0_6 2014

  5. [5]

    Castiglione & A

    G. Castiglione & A. Restivo (2005): Ordering and Convex Polyominoes. In: MCU 2004, Lecture Notes in Comput. Sci. 3354, Springer, pp. 128–139, doi:10.1007/978-3-540-31834-7_10

    work page doi:10.1007/978-3-540-31834-7_10 2005

  6. [6]

    Del Lungo, E

    A. Del Lungo, E. Duchi, A. Frosini & S. Rinaldi (2004): On the Generation and Enumeration of some Classes of Convex Polyominoes. Electron. J. Comb. 11(1), doi:10.37236/1813

    work page doi:10.37236/1813 2004

  7. [7]

    & Rogaway, P

    A. Del Lungo, A. Frosini & S. Rinaldi (2003): ECO Method and the Exhaustive Generation of Convex Polyominoes. In: DMTC 2003, Lecture Notes in Comput. Sci. 6795, Springer, pp. 129–140, doi:10.1007/3- 540-45066-1_10

    work page doi:10.1007/3- 2003

  8. [8]

    Duchi, S

    E. Duchi, S. Rinaldi & G. Schaeffer (2008): The number of Z-convex polyominoes. Adv. Appl. Math. 40(1), pp. 54 – 72, doi:10.1016/j.aam.2006.07.004

    work page doi:10.1016/j.aam.2006.07.004 2008

  9. [9]

    O’Rourke E

    J. O’Rourke E. D. Demaine, J. S. B. Mitchell: The Open Problems Project. Available at http://cs.smith. edu/~jorourke/TOPP

  10. [10]

    Formenti & P

    E. Formenti & P. Massazza (2017): From Tetris to polyominoes generation . Electronic Notes in Discrete Mathematics 59, pp. 79 – 98, doi:10.1016/j.endm.2017.05.007

    work page doi:10.1016/j.endm.2017.05.007 2017

  11. [11]

    Formenti & P

    E. Formenti & P. Massazza (2018): On the Generation of 2-Polyominoes. In: DCFS 2018, Lecture Notes in Comput. Sci. 10952, Springer, pp. 101–113, doi:10.1007/978-3-319-94631-3_9

    work page doi:10.1007/978-3-319-94631-3_9 2018

  12. [12]

    & Massazza, Paolo (2024): Asymptotics of Z-convex polyominoes

    Guttmann, Anthony J. & Massazza, Paolo (2024): Asymptotics of Z-convex polyominoes. RAIRO-Theor. Inf. Appl. 58, p. 26, doi:10.1051/ita/2024009

    work page doi:10.1051/ita/2024009 2024

  13. [13]

    D. A. Klarner, S.W. Golomb & G. Barequet (2017): Polyominoes. In J.E. Goodman, J. O’Rourke & C. D. Tóth, editors: Handbook of Discrete and Computational Geometry , CRC Press, Boca Raton, FL, pp. 359–

    2017

  14. [14]

    Available at https://www.csun.edu/~ctoth/Handbook/chap14.pdf

  15. [15]

    Del Lungo, M

    A. Del Lungo, M. Nivat, R. Pinzani & S. Rinaldi (2004): A bijection for the total area of parallelogram polyominoes. Discret. Appl. Math. 144(3), pp. 291–302, doi:10.1016/j.dam.2003.11.007

    work page doi:10.1016/j.dam.2003.11.007 2004

  16. [16]

    (2026): The On-Line Encyclopedia of Integer Sequences

    OEIS Foundation Inc. (2026): The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org

    2026

  17. [17]

    Tawbe & L

    K. Tawbe & L. Vuillon (2011): 2L-convex polyominoes: Geometrical aspects. Contributions Discret. Math. 6(1), doi:10.55016/ojs/cdm.v6i1.62040

    work page doi:10.55016/ojs/cdm.v6i1.62040 2011