Amendments from Version 1

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10.12688/f1000research.172910.2

Research Article

Articles

Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO

[version 2; peer review: 1 approved, 1 not approved]

Mohommed

Eman F.

Methodology Supervision Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-0030-6512 a 1 Abd Jassim

jalal

Funding Acquisition Software Writing – Review & Editing 1 1mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq

a emanfatel@uomustansiriyah.edu.iq

No competing interests were disclosed.

15 5 2026

2026

24 3 2026

2026

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies a class of polyominoes. The new class is defined through a representation on the James abacus through nested chain which denoted Ω -nested Abacus. Depended on partition, beta number, nested chain we defined the structural condition this class. A local transformations, called SSPT-transformation and MSPT-transformation are formulated on the even chains. Then, we drive explicit formulas for the number of position in each chain, number of even chain and the total number of all chains. The structure of new class lead to enumeration formulas for the objects produced after application new transformation. To enhance further, based on these classes, generating functions are also being formulated by employing enumeration of combinatorial objects (ECO). In ECO method, each object is obtained from smaller object by making some local expansions. These local expansions are described in a simple way by a succession rule which can be translated into a function equation for the generating function.

Polyominoes Class Enumeration Succession-rule Abacus diagram

The author(s) declared that no grants were involved in supporting this work.

Revised Amendments from Version 1

In this revised version, we have substantially improved the manuscript in response to the reviewers’ comments. Several definitions have been clarified, and all variables are now consistently defined and used throughout the paper. The statements and proofs of Lemma 11 and Theorem 12 have been revised to explicitly include all necessary assumptions and to ensure that the arguments are self-contained. In addition, Corollary 13 and Theorem 14 have been rewritten with unified notation and clearer combinatorial reasoning. The application of the ECO method has been reconstructed in greater detail, including a clearer description of the generating tree and the associated succession rule. Furthermore, the derivation of the generating function has been completely rewritten from first principles, with a more detailed and rigorous presentation. Finally, the overall language, structure, and readability of the manuscript have been carefully improved to enhance clarity and coherence. We believe that these revisions significantly strengthen the mathematical rigor and presentation of the work.

Introduction

Polyominoes are finite configurations composed of unit squares, known as ominoes, that are joined edge to edge to form a connected interior, as illustrated in Figure 1.

Figure 1. 22-polyominoes (polyominoes with 22 connected ominoes).

An n -polyomino (a polyomino consisting of n ominoes) is defined up to translation, and the concept is commonly attributed to Golomb. ¹ In contemporary research, n-polyominoes have attracted significant attention from computer scientists, physicists, mathematicians, and biologists. Despite extensive studies, enumerating n-polyominoes remains a challenging and unresolved problem in combinatorial geometry, representing one of its most fundamental open questions. ^{2–
5} There is no closed-form equation for n-polyominoes, and the problem has been solved up to n < 56. ³ No closed-form expression for the enumeration of n -polyominoes is currently known. Due to the complexity of this problem, several simpler subclasses of n -polyominoes have been formulated and extensively examined in the literature. ^{6–
8} E.F. provide a new representation of n-polyominoes (Plyominoes with n ominoe) using one of the graphical representations of partition (James-diagram) called nested chain abacus (N.C.A.) with b -columns and d -rows. ⁹ The Nested Chain Abacus (N.C.A.) offers a novel framework for representing any n connected omino, including empty ones (holes), through the use of a beta number. In this new representation, n -polyominoes are organised into a series of nested chains. ¹⁰ A Nested Chain Abacus (N.C.A.) is an n -polyomino inscribed within a James diagram, consisting of both outer and inner chains. The inner chains are numbered from 1 to n , where n is a positive integer, and chain 1 is the innermost chain. No intersections occur among any of the chains. Through this new representation, and for the first time, each n-polyomino has been systematically associated with a unique code. Next Figure 2 illustrates an example of an N.C.A. with six columns, five rows, and three chains.

Figure 2. Nested chain abacus with 6 columns, 5 rows and 3 chains. Terminologies and definition

This section introduced some importuned definitions: Definition 1.

⁴A partition of a positive integer, k , is a sequence of integers λ 1 , λ 2 , … , λ n such that λ 1 ≥ λ 2 ≥ … ≥ λ n and ∑ i = 1 n λ i = k .

Example 1.

λ = ( 6 , 6 , 3 , 2 , 1 , 1 ) is a partition of 19

Definition 2.

⁴A sequence of positive numbers { β 1 , β 2 , … , β k } called beta number such that β i = λ i + k − i and 1 ≤ i ≤ k .

Example 2.

⁴Consider of Example 1, λ = ( 6 , 6 , 3 , 2 , 1 , 1 ) then beta number of λ is β i = λ i + b − i , where i = 1 , 2 , 3 , 4 , 5 , 6

And hence β 1 = 11 , β 2 = 10 , β 3 = 6 , β 4 = 4 , β 5 = 2 , β 6 = 1

Thus beta number sequence is { 11 , 10 , 6 , 4 , 2 , 1 } .

Definition 3.

⁴James Abacus is a graphical representation of a partition of any integer number using beta numbers.

Consider Example 1, the James Abacus with 1,2,3,4,6,7,10,11 beta position as show in next Figure 3.

Definintion 4.

⁹A new representation of n-polyominoes (Plyominoes with n ominoe) using one of the graphical representations of partition (James-diagram) called nested chain abacus (N.C.A.) with b-columns and d-rows.

Next Figure 4 gives an example of N.C.A. with 3 chains represented of 94, where the beta number sequence is {0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Figure 3. James’s Abacus with 3 columns, 4 rows and 6 beta. Figure 4. N.C.A. with 5 columns ( <inline-formula> <mml:math display="inline"> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:math> </inline-formula>) and 3 chains represented of 94, where the beta number sequence is <inline-formula> <mml:math display="inline"> <mml:mo stretchy="true">{</mml:mo> </mml:math> </inline-formula>0,5,6,7,8,9,10,11,14,15,16,19,20,21,24}.

Definition 5:

A connected chain is a chain with only beta positions.

Next Figure 5 gives an example of N.C.A. represented by a beta sequence {0,1,2,3,4,5,7,8,9,10,12,14,15,19,20,21,22,23,24} with one connected chain (chain 1).

Figure 5. N.C.A. with 1 connected chains (chain 3). Definition 6.

Let β i , β j be two beta numbers in N.C.A. then β i , β j are connected iff 1-

| β i − β j | = 1 if β i , β j Located in the same row.

| β i − β j | = b if β i , β j Located in the same column .

For example, the beta set { 0 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 14 , 15 , 16 , 19 , 20 , 21 , 24 } .

From Figure 4 is a connected.

Definition 7.

Each nested chain abacus (N.C.A.) with b columns and b rows is called a polyamines if every two beta numbers are connected.

Definition 8.

Let N.C.A. be a nested chain abacus consist of b column and d rows, and let C 1 , C 2 , … , C r be its chains, order from the innermost chain to the outermost chain. The NCA is called an Ω -nested Abacus if the following condition are satisfied 1-

First chain (initial chain) C 1 which consist exactly one position. a.

If C 1 consist exactly one beta position, then C 2 has at least five beta positions.

If the position in C 1 is empty position, then C 2 consist k beta positions, where 1 ≤ k ≤ 8

Every even chain ( C 2 t ) may contain both beta position and empty beta position.

Every odd chain ( C 2 t − 1 ) forms a connected chain.

Any polyomino represented by such a configuration is called an Ω -nested Abacus polyomino.

Figure 6 gives an example of Ω -nested Abacus with four chains, where

Not Every polyamines inscribed in a nested chain-abacus is Ω -nested Abacus.

Figure 4 given an example of N.C.A. but not Ω -nested Abacus, while Figure 6 given an example of Ω -nested Abacus.

Not. Throughout our result the underlying new family ( Ω -nested Abacus) assumed b = d where b is odd number. Since every odd chain is completely filled, so only even chain well be contribute to the generating process.

The algorithm construct to generate classes of new family depended on the following transformation.

Figure 6. <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:math> </inline-formula>-nested Abacus with four chains.

Definition 9.

Single – beta transformation (SPT)

Let an Ω -nested Abacus with b column, d rows and C i chains. Suppose that a beta number β = ( m − 1 ) b + ( j − 1 ) in chain C i is located in row m and column j such that 1 ≤ m ≤ d , 1 ≤ j ≤ b and 1 ≤ i ≤ r . A single beta transformation (SPT) in chain C i is local transformation ( β → β ` ) within the same chain where β ` = { ( m − 1 ) b + ( j − 2 ) if i + 1 ≤ m ≤ ( r − i + 1 ) , j = b − i + 1 mb + ( j − 1 ) if i ≤ m ≤ ( r − i ) , j = i ( m − 1 ) b + j if m = i , i + 1 < j ≤ b − i + 1 ( m − 2 ) b + ( j − 1 ) if m = d − i + 1 , i ≤ j < b − i .

Remark 10.

The maximal number of transformations in chain n is equal to the number of positions in the chain, where n > 1 .

Figure 7 illustrates Single – beta transformation (SPT)

Not. 1-

SSpt is a SPT transformation in one chain.

MSpt is a SPT transformation in all chains.

Figure 7. Illustrates Single – beta transformation (SPT) on <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:math> </inline-formula>-nested Abacus. Enumeration of the <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold">Ω</mml:mi> </mml:math> </inline-formula>-nested Abacus class

Next, the enumeration of Ω-nested Abacus with b columns and d rows is presented. Lemma 11.

Let an Ω -nested Abacus with b column, d rows and C r chains then, the number of position in any chains is | C i | = 8 ( n − 1 ) .

Proof.

Based on Definition 9 C 1 consists of a single position. Then, the successive chain form a discrete ring around the single position with four side increments and four corner position. Then | C 2 | = 8 , thus | C n | = | C n − 1 | + 8 where 1 ≤ n ≤ r . Thus | C n | = 8 ( n − 1 )

Where n > 1 .

Corollary 12.

Based on Remark 10 and Lemma 11, the number of (SSPT) transformation in chain C n is equal to 8 ( n − 1 ) .

Lemma 13.

Let Ω-nested Abacus be a polyomino intercept in N.C.A. with b columns and d rows such that b = d and b is odd. Then the total number of chain is b + 1 2 .

Proof.

Based on Ω-nested Abacus structure is symmetric with respect to a central column (central chain C n ) then for all i-th chains, the boundary columns must satisfy i ≤ b − i + 1 , thus

2 i ≤ b − 1 → i ≤ b + 1 2 . Then every Ω-nested Abacus with b + 1 2 chains.

Proposition 14.

Let Ω-nested Abacus be a polyomino intercept in NCA with b columns and d rows such that b = d and b is odd. Then the number of even chain is b − 1 4 .

Provided that b ≡ 1 ( mod 4 ) .

Proof.

Based on Lemma 13 the number of chain in Ω-nested Abacus is t = b + 1 2 . The even chains among 1 , 2 , … , t are 2 , 4 , … , ⌊ t 2 ⌋ . Since t = b + 1 2 is even then t 2 = b + 1 2 2 is even since the first chain is odd and fixed. Then there are Since t = b + 1 2 is even then t 2 = b + 1 2 − 1 2 = b − 1 4 , then the number of even chain is b − 1 4 .

Next we found the number of Ω-nested Abacus if we application SSPT-Transformation in chain i .

Lemma 15:

Let Ω-nested Abacus be a polyomino intercept in NCA with b columns and d rows such that b = d , b is odd and let n be even number. Then the number of Ω-nested Abacus generating by employ SSPT-Transformation exactly chain n is 8 ( n − 1 ) .

Proof.

By Lemma 11 the number of admissible positions in chains n is 8 ( n − 1 ) . Since move each beta position yields a distinct Ω-nested Abacus under SSPT-transformation obtain in chain n with 8 ( n − 1 ) position is 8 ( n − 1 ) .

Theorem 16:

Let Ω-nested Abacus be a polyomino intercept in N.C.A. with b columns and d rows such that b = d , b is odd and let n be even number. Then the number of Ω-nested Abacus generating by employ SSPT-Transformation is ∑ ∀ n ( R ( n − 1 ) 8 )

Where R = ∑ n ⌊ b + 1 4 ⌋ ( n − 1 ) 8 , b ≡ 1 ( mod 4 )

Proof.

Based on Lemma 10, the number of positions in chain n is ( n − 1 ) 2 3 by Lemma 11 and Definition 8, there are b − 1 4 chain with beta and empty beta position in Ω-nested Abacus. The maximal number of transformations in chain n is equal to ( ( n − 1 ) 8 ) . Since each move generates a new Abacus (polyominoes), thus there are ( ( n − 1 ) 8 ) of Ω-nested Abacus generated by employing SSPt transformation. As a result of this, there are ∑ ∀ n ( R ( n − 1 ) 8 )

Ω-nested Abacus.

Theorem 17.

Where b ≡ 1 ( mod 4 )

Proof.

Based on Lemma 11 the number of positions in chain n is 8( n−1), any position in the chain will be determines one distinct SSPT in that chain. Thus all even chain C 2 , C 4 , … , gives ∑ n = 2 ⌊ b + 1 4 ⌋ 8 ( n − 1 ) .

Theorem 18.

Let Ω-nested Abacus be a polyomino intercept in N.C.A. with b columns and d rows such that b = d , b is odd and let n be even number. Then the number of Ω-nested Abacus generating by employ MSPT-Transformation exactly chain n is ∏ ∀ n 8 ( n − 1 )

Where b ≡ 1 ( mod 4 ) and n = 2 , 4 , … . . , b = 1 4

Proof.

Based on Lemma 11, the number of admissible beta and empty eta position in chain n is 8 ( n − 1 ) . Since the odd chain are completely filled, the remain fixed and do not contribute to the transformation. By Lemma 15 the number of Ω-nested Abacus generating by employ SSPT-Transformation exactly chain n is 8 ( n − 1 ) . Since the MSPT-transformation acts simultaneous on all even chain. A generated Ω-nested Abacus is obtain by selecting one admissible position in each chain. The choice in one even chain does not restrict the admissible choices in any other even chain. Then the number of Ω-nested Abacus can be generated after application SSPT-Transformation is ∏ ∀ n 8 ( n − 1 ) .

Generating function with respect to chains

In this section, the method described in ^{11,
12} is employed to enumerate the Ω-nested Abacus class, representing polyominoes inscribed within a James diagram. The ECO (Enumerating Combinatorial Objects) method has previously been applied to the enumeration of various polyomino classes. ¹² This approach is based on a succession rule.

Generating Function (G.F.)

Let C 2 k be even chain of Ω-nested Abacus, based on Lemma 11 the number of positions in any chain is M k = ⌈ C 2 k ⌉ = 8 ( n − 1 ) .

We application SSPT-Transformation inside this chains such that no previously select position may be chose again. Thus, in initial stage there are M k choices ( M k of Ω-nested Abacus), after one insertion we have only M k − 1 position ( M k − 1 of Ω-nested Abacus), after two distinct insertion there are only M k − 2 position ( M k − 2 of Ω-nested Abacus as shown in Figure 8), and so on. Thus, there are a n k = M k ( M k − 1 ) ( M k − 2 ) … ( M k − n + 1 ) , 2 ≤ n ≤ M k

Figure 8. First and second level of <inline-formula> <mml:math display="inline"> <mml:mi>ϑ</mml:mi> </mml:math> </inline-formula> using <inline-formula> <mml:math display="inline"> <mml:mi>Ω</mml:mi> </mml:math> </inline-formula>-nested Abacus class with 5 columns, 5 rows and 3 chain.

Above process is encoded by a generating tree. The root labeled by M k , each node will be produces j children, every one of j children will be product j − 1 . Thus the local succession rule is ϑ = { M k j → ( j − 1 ) j . 1 ≤ j ≤ M k

Assume that a n + 1 k denoted the number of nodes at level n ϑ yield the recurrence a n + 1 k = ( M k − n ) a n k

Where a 0 k = 1 , 2 ≤ n ≤ M k . a n k = M k ! ( M k − n ) ! .

Using the ordinary level-generating polynomial of the even chain C k ( x ) = ∑ n = 2 M k ( M k ) n x n C k ( x ) = ∑ n = 2 M k M k n n ! x n

Since ( M k ) n n ! = M k ! ( M k − n ) ! n ! = ( M k n ) then, C k ( x ) = ∑ n = 2 M k ( M k n ) x n = ( 1 + x ) M k C k ( x ) = ∑ n = 2 M k ( M k n ) x n = ( 1 + x ) 8 ( 2 k − 1 )

Then the generating function C k ( z , x ) = ∑ k = 2 m z k ( C k ( x ) ) = ∑ n = 2 M k z k ( 1 + x ) 8 ( 2 k − 1 )

Where m = ⌊ b + 1 4 ⌋

Thus C k ( z , x ) = ∑ n = 2 M k ( z ( 1 + x ) 16 ) k C k ( z , x ) = z ( 1 + x ) 8 1 − ( z ( 1 + x ) 16 ) m 1 − z ( 1 + x ) 16

C k ( z , x ) is generating function to enumerated the number of Ω-nested Abacus.

Conclusion

This study examines a class of polyominoes known as the Ω-nested Abacus. Initially, we provide a characterization of a new class based on specific geometric constraints defined by rows, columns, and chains. A series of operations on polyominoes is then introduced using a partition-theoretic construct known as the beta number. Furthermore, a set of operations on polyominoes is developed using a partition-theoretic construct known as the beta number, allowing for localized transformation of the structure. Furthermore, A succession rule is formulated based on generating tree techniques to describe the growth of these object. Based on this framework a recursive method is established for the systematic generation of Ω-nested Abacus configuration of a given size through the use of generating trees.

Ethical approval

Ethical approval was not required for this study.

Data availability statement

No data availability with this atrial as the study is purely theoretical and does not involve the generating analysis or use of any datasets.

References 1

Golomb

: Checkerboards and polyominoes. Am. Math. Mon. 1954;61(10):675682. 10.1080/00029890.1954.11988548

Flórez

José Ramírez

Villamizar

: Restricted bargraphs and unimodal compositions. Journal of Combinatorial Theory, Series A. 2024;208:105934. 10.1016/j.jcta.2024.105934

Arabi

Belbachir

Dubernard

: Enumeration of plateau polycubes with respect to their lateral area. Indian Journal of Pure and Applied Mathematics. 2024;55:538–554. 10.1007/s13226-023-00385-3

Jensen

: Counting polyominoes: a parallel implementation for cluster Computing. Proceedings of the 2003 International Conference on Computational Science: Part III, ICCS’03. Springer-Verlag;2003; pp.203–212.

Castiglione

Restivo

: Ordering and convex polyominoes. the International Conference on Machines, Computations, and Universality. Berlin, Heidelberg: Springer Berlin Heidelberg;September 2004; pp.128–139.

Del Lungo

: Polyominoes defined by two vectors. Theor. Comput. Sci. 1994;127(1):187–198. 10.1016/0304-3975(94)90107-4

Castiglione

Frosini

Restivo

: A tomographical characterisation of convex polyominoes. Poitiers, Proc. of Discrete Geometry for Computer Imagery 12th International Conference, DGCI 2005, in: Lecture Notes in Computer. 2005; pp.115–125.

Castiglione

Restivo

: Reconstruction of L-convex polyominoes. Electron. Notes Discret. Math. 2003;12:290–301. 10.1016/S1571-0653(04)00494-9

Mohommed

: Topological Structure of Nested Chain Abacus. Iraqi Journal of Science. 2020;153–160. 10.24996/ijs.2020.SI.1.20

Mohommed

: Constructing a Nested Chain in James Abacus Diagram. J. Phys. Conf. Ser. 2019;1294(3):032019. IOP Publishing. 10.1088/1742-6596/1294/3/032019

grammars to ECO systems.

Theor. Comput. Sci. 2004;57–95.

Banderier

Wallner

: Lattice paths with catastrophes. Discrete Mathematics & Theoretical Computer Science. 2017;19(1). Analysis of Algorithms. 10.23638/DMTCS-19-1-23

10.5256/f1000research.197917.r485075

Reviewer response for version 2

Hashim

Hayder R.

1 Referee https://orcid.org/0000-0001-5408-7496 1University of Kufa, Kufa, Iraq

Competing interests: No competing interests were disclosed.

25 5 2026

2026

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve

The authors have revised the manuscript in accordance with my comments. My only remaining suggestion to the authors is to carefully polish the language. Otherwise, I recommend acceptance for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

Number Theory and cryptography

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

10.5256/f1000research.190675.r462385

Reviewer response for version 1

Hashim

Hayder R.

1 Referee https://orcid.org/0000-0001-5408-7496 1University of Kufa, Kufa, Iraq

Competing interests: No competing interests were disclosed.

5 3 2026

2026

recommendation

approve-with-reservations

The manuscript proposes a new class of polyominoes defined by a graphical representation referred to as the James Abacus or Nested Chain Abacus and attempts to enumerate this class using combinatorial transformations and the ECO method.

Since the results in the paper are interesting in number theory, I suggest accepting the paper after revising it according to the following remarks as the paper would benefit from improvements in clarity, organization, and presentation.

1. Lemma 11 and Theorem 12 refer to external theorems without clearly stating assumptions or applicability.

2. Corollary 13 and Theorem 14 contain inconsistent notation such as mixing b,p,r,e,d and unclear counting logic.

3. Polish the language carefully.

4. Every lemma and theorem must include a complete, self-contained proof.

5. All variables must be clearly defined and used consistently.

6. If results rely on previous publications, the exact statements must be cited and restated.

7. Try to reconstruct in more details the ECO method application properly.

8. Redo or rewrite the derivation of the generating function theorem from first principles, with a more clear form.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

Number Theory

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Mohommed

Eman

mathematics, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq

Competing interests: No competing interests were disclosed.

19 3 2026

We thank the reviewer for his comments

Comment 1:

Lemma 11 and theorem 12 refer to external theorems without clearly stating assumptions of applicability

Response: in the revised version, we have explicitly stated all required assumptions and clarified the applicability of the referenced result. In addition, we have added several lemmas and theorems related with lemma 11 and theorem 12 to ensure that lemma 11 and theorem 12 are fully self-contained and mathematically precise.

Comment 2:

Corollary 13 and Theorem 14 contain inconsistent notation, such as mixing b, p, r, e, d, and unclear counting logic

Response: we appreciate this observation. The notation has been carefully revised and unified throughout the paper. All variables are now clearly defined and used consistently.

Comment 3:

Polish the language carefully

Response: all manuscripts have been proofread and language has been improved.

Comment 4:

Every lemma and theorem must include a complete, self-contained proof.

Response: In the new version, all lemmas and theorems now include complete proof. Additional explanations have been added where necessary.

Comment 5:

All variables must be clearly defined and used consistently.

Response:

All variables have been systematically reviewed and defined at their first occurrence and then has been standardized across the paper to ensure consistency.

Comment 6:

If results rely on previous publications, the exact statements must be cited and restated

Response: we revised all relevant sections to explicitly cite the original source.

Comment 7,8:

Try to reconstruct in more detail the proper application of the ECO method.

Redo or rewrite the derivation of the generating function theorem from first.

Response: following the reviewer's suggestion, we have completely rewritten the derivation of the generating function.

10.5256/f1000research.190675.r450050

Reviewer response for version 1

Qureshi

Ayesha

1 Referee 1Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Istanbul, Turkey

Competing interests: No competing interests were disclosed.

16 1 2026

2026

recommendation

reject

This paper introduces a class of polyominoes defined via a nested chain abacus representation and aims to enumerate this class using transformations and the ECO method. While the topic may be of interest, the manuscript in its current form lacks sufficient mathematical clarity and rigor.

Several definitions are unclear or inconsistent, and many results are stated without complete or verifiable proofs. In particular, the use of the ECO method and the derivation of the generating function require a more precise and rigorous formulation. Additionally, the presentation and language need substantial improvement. A major revision addressing these issues would be necessary before the conclusions can be considered supported by the results.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

Algebra

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.