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Mohommed\" }, { \"@type\": \"Person\", \"name\": \"jalal Abd Jassim\" } ], \"publisher\": { \"@type\": \"Organization\", \"name\": \"F1000Research\", \"logo\": { \"@type\": \"ImageObject\", \"url\": \"https://f1000research.com/img/AMP/F1000Research_image.png\", \"height\": 480, \"width\": 60 } }, \"image\": { \"@type\": \"ImageObject\", \"url\": \"https://f1000research.com/img/AMP/F1000Research_image.png\", \"height\": 1200, \"width\": 150 }, \"description\": \"This paper studies a new family of polyominoes. The family is defined through a graphic representation called James diagram, also known as the nested chain abacus. This form is referred to as the Ω-nested Abacus. The proposed class is defined by geometric constraints satisfied by sets of rows, columns, and chains. Through the application of combinatorial approaches and variable generating function, we enumeration of this new class of polyominoes. During the new construction process transformation is formulates which plays a crucial role in the enumerate of a family of classes. To further extend this study generating function for theses class are formulated using an enumeration method known as the Enumeration Combinatorial Object (ECO) technique. ECO method, derived from a smaller one though a series of local expansion. Theses expansion are systematically defined by succession rule which can subsequently be expressed as a functional equation for the generating function.\" } { \"@context\": \"http://schema.org\", \"@type\": \"BreadcrumbList\", \"itemListElement\": [ { \"@type\": \"ListItem\", \"position\": \"1\", \"item\": { \"@id\": \"https://f1000research.com/\", \"name\": \"Home\" } }, { \"@type\": \"ListItem\", \"position\": \"2\", \"item\": { \"@id\": \"https://f1000research.com/browse/articles\", \"name\": \"Browse\" } }, { \"@type\": \"ListItem\", \"position\": \"3\", \"item\": { \"@id\": \"https://f1000research.com/articles/15-11/v1\", \"name\": \"Enumeration Class of Polyominoes Inscribed in James Abacus and Related...\" } } ] } Home Browse Enumeration Class of Polyominoes Inscribed in James Abacus and Related... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Mohommed EF and Abd Jassim j. Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.12688/f1000research.172910.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] Eman F. Mohommed https://orcid.org/0000-0003-0030-6512 1 , jalal Abd Jassim 1 Eman F. Mohommed https://orcid.org/0000-0003-0030-6512 1 , jalal Abd Jassim 1 PUBLISHED 06 Jan 2026 Author details Author details 1 mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq Eman F. Mohommed Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing jalal Abd Jassim Roles: Funding Acquisition, Software, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract This paper studies a new family of polyominoes. The family is defined through a graphic representation called James diagram, also known as the nested chain abacus. This form is referred to as the Ω-nested Abacus. The proposed class is defined by geometric constraints satisfied by sets of rows, columns, and chains. Through the application of combinatorial approaches and variable generating function, we enumeration of this new class of polyominoes. During the new construction process transformation is formulates which plays a crucial role in the enumerate of a family of classes. To further extend this study generating function for theses class are formulated using an enumeration method known as the Enumeration Combinatorial Object (ECO) technique. ECO method, derived from a smaller one though a series of local expansion. Theses expansion are systematically defined by succession rule which can subsequently be expressed as a functional equation for the generating function. READ ALL READ LESS Keywords Polyominoes, Class, Enumeration, Succession-rule, Abacus diagram Corresponding Author(s) Eman F. Mohommed ( [email protected] ) Close Corresponding author: Eman F. Mohommed Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 Mohommed EF and Abd Jassim j. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Mohommed EF and Abd Jassim j. Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.12688/f1000research.172910.1 ) First published: 06 Jan 2026, 15 :11 ( https://doi.org/10.12688/f1000research.172910.1 ) Latest published: 15 May 2026, 15 :11 ( https://doi.org/10.12688/f1000research.172910.2 )  There is a newer version of this article available. Suppress this message for one day. 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. A partition of any 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. A 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 . λ Example 2. Consider of Example 1 , λ = ( 4 , 4 , 2 , 2 , 1 , 1 , 1 , 1 ) then beta number of λ is β i = λ i + b − i , where i = 1 , 2 , 3 , 4 , 5 , 6 β 1 = λ 1 + b − 1 = 4 + 8 − 1 = 11 β 2 = λ 2 + b − 2 = 4 + 8 − 2 = 10 β 3 = λ 3 + b − 3 = 2 + 8 − 3 = 7 β 4 = λ 4 + b − 4 = 2 + 8 − 4 = 6 β 5 = λ 5 + b − 5 = 1 + 8 − 5 = 24 β 6 = λ 6 + b − 6 = 1 + 8 − 6 = 3 β 7 = λ 7 + b − 7 = 1 + 8 − 7 = 2 β 8 = λ 8 + b − 8 = 1 + 8 − 8 = 1 Thus beta number sequence is { 1 , 2 , 3 , 4 , 6 , 7 , 10 , 11 } . Definition 3. James 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 . 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 ( b = 5 ) and 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}. Definition 4: 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 5. Let β i , β j be two beta numbers in N.C.A then β i , β j are connected iff 4 1- | β i − β j | = 1 if β i , β j Located in the same row. 2- | β i − β j | = b if β i , β j Located in the same column . Consider Figure 4 ; the set of beta {0,5,6,7,8,9,10,11,14,15,16,19,20,21,24} is a connected beta set. Definition 6. Each nested chain abacus (N.C.A) with b columns is called a polyamines if every two beta numbers are connected. Definition 7. Ω - nested Abacus is a N.C.A with b columns and d rows satisfy the following condition. 1- chain 1 with only one position a) If chain 1 has a beta position, then chain 2 has at least 5 beta positions. b) If chain 1 with empty beta position, then chain 2 has k beta positions, where 1 ≤ k ≤ 8 2- Chain number o are connected chain where o is an odd number. 3- Chain number e contains beta position as well as empty beta where o is an odd number. 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. Figure 6. Ω - nested Abacus with four chains. Definition 8. Single – beta transformation (SPT) is a beta movement in chain i is ⟶ β ` = { ( m − 1 ) b + ( j − 1 ) } { β ` = ( m − 1 ) b + ( j − 2 ) if i + 1 ≤ m ≤ ( r − i + 1 ) , j = e − i + 1 β ` = mb + ( j − 1 ) if i ≤ m ≤ ( r − i ) , j = i β ` = ( m − 1 ) b + j if m = i , i + 1 < j ≤ e − i + 1 β ` = ( m − 2 ) b + ( j − 1 ) if m = r − i + 1 , i ≤ j < e − i . With d rows, b columns where β = { ( m − 1 ) b + ( j − 1 ) } is the beta number in column number j and row number m . 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. 2- MSpt is a SPT transformation in all chains. Figure 7. Illustrates Single – beta transformation (SPT) on Ω - nested Abacus. Enumeration of the Ω - nested Abacus class Next, the enumeration of Ω-nested Abacus with b columns and d rows is presented. Lemma 10. Let of Ω-nested Abacus be Abacus with b rows and b columns, then the number of positions in chain n is ( ( n − 1 ) 2 3 ) where n > 1 and b is an odd number. Proof. Based on, 8 chain 1 consists of one position. To build the second chain, we add a position above, below, to the right, and to the left of chain number 1(consisting of one position), as well as the far right and far left from the top and the far right and left from the bottom of chain number 1. Then the number of positions in chain 2 is equal to 2 3 . Based on theorem 2.5.11, 13 the difference between chain i and chain ( i + 1 ) = 2 3 . Thus, the number of positions in chain n is ( ( n − 1 ) 2 3 ) where n > 1 . Lemma 11. The number of chains in the Ω-nested Abacus is b + 1 2 where b is the number of columns in Ω-nested Abacus. Proof. See Theorem 2.5.21. 13 Theorem 12: Let a be the number of even chains, then the number of Ω-nested Abacus generating by employ MSPT- Transformation is: ∑ ( b + 1 4 ( n − 1 ) 2 3 ) 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 p + 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 ) 2 3 ) . Since each move generates a new Abacus (polyominoes), thus there are ( ( n − 1 ) 2 3 ) of Ω-nested Abacus generated by employing MSpt transformation. As a result of this, there are ( b + 1 4 ( n − 1 ) 2 3 ) Ω--nested Abacus in chain n . Corollary 13: The number Ω-nested Abacus with b columns and d rows in chain i is 2 d + 2 b – 4 ( 2 i – 1 ) Proof. Based on theorem 2.5.9 13 Chain i have 2 d + 2 b − 4 ( 2 i – 1 ) position then the each beta can be move 2 d + 2 b – 4 ( 2 i – 1 ) so 2 d + 2 b – 4 ( 2 i – 1 ) shapes can be construct after application MSPT. Theorem 14: The number Ω-nested Abacus after the application MSPT transformation: π ( p + 1 4 ) 2 d + 2 p _ 4 ( 2 i − 1 ) proof. Based on (Corollary 1), then are 2 r + 2 e – 4 ( 2 i – 1 ) position in chain i since we have p + 1 4 chain with empty beta, then the number Ω-nested Abacus is π ( p + 1 4 ) 2 d + 2 p ( 2 i − 1 ) 4 . 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. 9 , 10 This approach is based on a succession rule. Succession rule It relies on a well-defined notion of size, such that the number of objects of size n is finite for any positive integer n. There exists exactly one object of size 1. A generating tree for any class is an infinite rooted tree whose vertices are the objects of the class, each appearing exactly once in the tree, and such that objects of size n are at level n . If we give a class ϑ of discrete objects and a parameter ρ on ϑ , where ϑ n = { x ∈ ϑ : p ( x ) = n } . Let θ be an operator which constructs every object Y ∈ ϑ n from another Y ∈ ϑ n such that Y get from exactly one X ∈ ϑ n − 1 . then, we have a recursive construction of all the elements of Ω ; from this, in turn, we will some of the time deduce a functional equation verified by the creating work of ϑ . The development can be represented through a growth tree, where vertices corresponding to objects that share the same measurement with respect to the parameter p are positioned at the same level. The children of a given vertex correspond to the objects derived from it, and each vertex is labeled according to the number of its offspring. This values the richness of the nodes (and of the comparing object). If the names of the children of a hub named ( k ), as it were, depend on the k , able to speak to the developing stage of the tree by implies of the following (called succession-rule). ϑ = { ( X ) ( Y ) → ( y 1 ) ( y 2 ) … ( y i ) Where X is the tag of the root and ( y i ) is the tag of the k-th son of a node tagged i , a succession-rule can be represented by a generating tree. The purpose of this section is to demonstrate how variations in succession rules can influence the structure of the corresponding generating function. Moreover, describe the class of Ω-nested Abacus polyominoes, which are enumerated by factorial numbers with respect to their coordinated height. Furthermore, examine the succession rule that represents the growth pattern of this class as derived using the ECO method. From the succession rule, it produces a sequence, { g n } n , of positive integers, such that g n is the number of nodes in level n of the generating tree, and the generating function of the generating tree is denoted by g ϑ = ∑ g n x n (Bacchelliet al., 2010). In this work, our succession rule will be constructed starting from a single chain (chain with a beta), which will grow step by step by adding C i beads, where C i is the number of positions in chain i, as shown in Figure 4.10, which illustrates the number of nested chain abacuses in levels 1 and 2. A label is assigned containing all the essential information necessary to describe how the Ω-nested Abacus class evolves within the generating tree. Generating tree • First level of the generating tree of ϑ begins with Ω - nested Abacus class consist of a single beta ( β s ) where ( β s = b 2 − 1 2 ) . • Second level L 2 we add one of the following beta number sets called β L 2 where β L 2 = { β s + 1 , β s − 1 , β s + b , β s − b , β s + b + 1 , β s + b − 1 , β s − b + 1 , β s − b − 1 } Such that b is the number of columns, so the 8 of Ω - nested Abacus class called Ω L 2 . • Third level L 3 in this level e take one of Ω L 2 class add one of the β L 2 set, so the 7 of Ω - nested Abacus class called Ω L 3 and then, as shown in Figure 8 and Figure 9 . Figure 8. β L 2 set in Ω - nested Abacus class. Figure 9. First and second level of ϑ using Ω - nested Abacus class with 5 columns, 5 rows and 3 chain. The succession rule of Ω -nested Abacus (polyominoes) Let Ω - nested Abacus be a polyominoes class with b columns and d rows, where i be a number of chains. The initial set began with two chains and a singleton beta ( β s ) . The Ω - nested Abacus class can be described, depending on the generating tree, by the overview rule: ϑ = { ( 1 ) ( 8 i − 8 ) → ( 1 ) ( 2 ) … ( 8 i − 8 ) ! . Lemma 15: Let i be the number of chains. Then, the number of Ω -nested Abacus in the generating tree by adding a beta number in chain i at level n is { L O = 1 L 1 = 8 i − 8 L 2 = 8 i − 7 L 3 = 8 i − 6 ⋮ ⋮ L n = ( 8 i − 1 ) ! . . . … … … ( ∗ ) Generating Function (G.F) Theorem. Let In order to define G.F., first we have to find a succession rule, ϑ . Which is ϑ = { ( 1 ) ( 8 i − 8 ) → ( 1 ) ( 2 ) … ( 8 i − 8 ) ! . Then we must find the function. g n ( x , y ) and g n ( x ) when y = 1 to find the generating function. We can define a sequence of positive integers. g n , n ≥ 0 , where g n The number of nodes at the general level (level n). In the succession-rule, ϑ (*), all nodes are changed to (8i-8). Then: g n ( x , y ) = ∑ n ≥ 0 i ≥ 0 g n , 8 ( i − 1 ) ∂ g ϑ ( x , y ) ∂ y − 1 x y 8 g ϑ ( x , y ) = − y 8 i − 8 − 8 x g ϑ ( x , y ) = e ∫ − x y 8 dy ( c + ∫ − y 8 i − 8 − 8 x e ∫ 1 x y 8 dy dy ) = 1 e 7 x y 7 ( c − ∫ y 8 ( i − 1 ) e 1 − 7 x y 7 dy x ) lim x ⟼ 0 g ϑ ( x , 1 ) = lim x ⟼ 0 1 e 7 x ( c − lim x ⟼ 0 e 1 − 7 x x ) e 8 ( c − 0 ) = 0 Then c = 0 g ϑ ( x , y ) = 1 e 7 x y 7 ( − ∫ y 8 i e 1 − 7 x y 7 dy x ) Conclusion This study examines a class of polyominoes known as the Ω-nested Abacus. Initially, we study the characterised of a new class by specific geometric constraints defined by sets of 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, enabling localised transformations of the objects. Furthermore, A succession rule is formulated based on tree structures that illustrate the developmental patterns of these trees. A recursive method is established for the systematic generation of these objects 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 S: Checkerboards and polyominoes. Am. Math. Mon. 1954; 61 (10): 675682. Publisher Full Text 2. Flórez R, José Ramírez L, Villamizar D: Restricted bargraphs and unimodal compositions. Journal of Combinatorial Theory, Series A. 2024; 208 : 105934. Publisher Full Text 3. Arabi A, Belbachir H, Dubernard JP: Enumeration of plateau polycubes with respect to their lateral area. Indian Journal of Pure and Applied Mathematics. 2024; 55 : 538–554. Publisher Full Text 4. Jensen C: 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. 5. Castiglione G, Restivo A: Ordering and convex polyominoes. the International Conference on Machines, Computations, and Universality. Berlin, Heidelberg: Springer Berlin Heidelberg; September 2004; pp. 128–139. 6. Del Lungo A: Polyominoes defined by two vectors. Theor. Comput. Sci. 1994; 127 (1): 187–198. Publisher Full Text 7. Castiglione G, Frosini A, Restivo A, et al. : 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. 8. Castiglione G, Restivo A: Reconstruction of L-convex polyominoes. Electron. Notes Discret. Math. 2003; 12 : 290–301. Publisher Full Text 9. Mohommed EF: Topological Structure of Nested Chain Abacus. Iraqi Journal of Science. 2020; 153–160. Publisher Full Text 10. Mohommed EF: Constructing a Nested Chain in James Abacus Diagram. J. Phys. Conf. Ser. 2019; Vol. 1294 (3): 032019. IOP Publishing. Publisher Full Text 11. grammars to ECO systems. Theor. Comput. Sci. 2004; 57–95. 12. Banderier C, Wallner M: Lattice paths with catastrophes. Discrete Mathematics & Theoretical Computer Science. 2017; 19 (1). Analysis of Algorithms. Publisher Full Text 13. Mohommed E: A family of classes in nested chain abacus and related generating functions. Universiti Utara Malaysia; 2017. Diss. Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 06 Jan 2026 ADD YOUR COMMENT Comment Author details Author details 1 mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Baghdad Governorate, Iraq Eman F. Mohommed Roles: Methodology, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing jalal Abd Jassim Roles: Funding Acquisition, Software, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (2) version 2 Revised Published: 15 May 2026, 15:11 https://doi.org/10.12688/f1000research.172910.2 version 1 Published: 06 Jan 2026, 15:11 https://doi.org/10.12688/f1000research.172910.1 Copyright © 2026 Mohommed EF and Abd Jassim j. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Mohommed EF and Abd Jassim j. Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.12688/f1000research.172910.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 06 Jan 2026 Views 0 Cite How to cite this report: Hashim HR. Reviewer Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r462385 ) The direct URL for this report is: https://f1000research.com/articles/15-11/v1#referee-response-462385 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 05 Mar 2026 Hayder R. Hashim , University of Kufa, Kufa, Iraq Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190675.r462385 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 ... Continue reading READ ALL 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Hashim HR. Reviewer Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r462385 ) The direct URL for this report is: https://f1000research.com/articles/15-11/v1#referee-response-462385 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 15 May 2026 Eman Mohommed , mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq 15 May 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 15 May 2026 Eman Mohommed , mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq 15 May 2026 Author Response 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 ... Continue reading 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. 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. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Qureshi A. Reviewer Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r450050 ) The direct URL for this report is: https://f1000research.com/articles/15-11/v1#referee-response-450050 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 16 Jan 2026 Ayesha Qureshi , Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Istanbul, Turkey Not Approved VIEWS 0 https://doi.org/10.5256/f1000research.190675.r450050 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 ... Continue reading READ ALL 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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 Are the conclusions drawn adequately supported by the results? No Competing Interests: No competing interests were disclosed. 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Qureshi A. Reviewer Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r450050 ) The direct URL for this report is: https://f1000research.com/articles/15-11/v1#referee-response-450050 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 06 Jan 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 2 (revision) 15 May 26 Version 1 06 Jan 26 read read Ayesha Qureshi , Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Turkey Hayder R. Hashim , University of Kufa, Kufa, Iraq Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Hashim H. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 05 Mar 2026 | for Version 1 Hayder R. Hashim , University of Kufa, Kufa, Iraq 0 Views copyright © 2026 Hashim H. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (1) Author Response 15 May 2026 Eman Mohommed, mathematic, Mustansiriyah University Department of Mathematics, Baghdad, Iraq 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. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Hashim HR. Peer Review Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r462385) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-11/v1#referee-response-462385 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Qureshi A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 16 Jan 2026 | for Version 1 Ayesha Qureshi , Sabanci Universitesi (Ringgold ID: 52991), Istanbul, Istanbul, Turkey 0 Views copyright © 2026 Qureshi A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Not Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions 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 Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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 Are the conclusions drawn adequately supported by the results? No Competing Interests No competing interests were disclosed. 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. reply Respond to this report Responses (0) Qureshi A. Peer Review Report For: Enumeration Class of Polyominoes Inscribed in James Abacus and Related ECO [version 1; peer review: 1 approved with reservations, 1 not approved] . F1000Research 2026, 15 :11 ( https://doi.org/10.5256/f1000research.190675.r450050) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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