Topological Structure of Nested Chain Abacus

This paper focuses on developing a strategy to represent the -connected ominoes using an abacus. We use the idea of -connected ominoes with respect to a frame in modelling nested chain abacus. Then, we formulate and prove the unique connected partition for any -connected ominoes. Next, the topological structure of nested chain abacus is presented.


Definition 2.1
The -connected ominoes is a plane geometric figure formed by one or more ominoes (equal squares), such that there exists a path from one ominoes to another for any pair of ominoes which are internally connected and have holes. Next, we define the graphical form of -connected ominoes with respect to a minimal frame, which enables us to define ominoe positions and empty ominoe positions in terms of the rows and columns of the minimal frame.

Definition 2.2
A minimal frame with rows and columns for any -connected ominoes is such that there is at least one ominoes in each column and each row where . Consider Figure-1 where the minimal frame of 7-connected ominoes is as illustrated in Figure-

Definition 3
Any -connected ominoes in a minimal frame of can be represented in a nested chain abacus with beads where each bead position is obtained using the function , such that, if location in the minimal frame contains an ominoes, then for where and refer to the number of the rows and columns of the minimal frame, respectively. A nested chain abacus is an abacus with columns, rows, chains and connected bead positions, which satisfy the following conditions: Every bead position will be represented by a circle ( ) (Figure-2) , while the empty bead position will be represented by a dashed circle.
A connected partition is a sequence of non-increasing positive integer numbers ( ) such that represents connected bead positions and ∑ where is the number of parts in the connected partition. Next, we will construct an algorithm for the connectedness of bead positions.

2.Nested chain abacus
A new representation of -connected ominoes, called nested chain abacus, has been constructed. We use to explain the algorithm for nested chain abacus.
Step 1: Establishing a graphical form of -connected ominoes Consider Figure-  Step 2: Creating a direction of -connected ominoes w.r.t minimal frame In this step, we created a direction to obtain nested chain abacus with respect to the minimal frame. We begin with the top-leftmost positions, from left to right, and from top to bottom in the minimal frame. Consider Figure-1.b where we begin from bead position A which is located in row 1 and column 1, then bead position B in row 1 and column 2. The third, fourth,... positions for the minimal frame for -connected ominoes are substituted with the remaining positions C, D, E, F, G, respectively. Subsequently, we can also observe that there are two empty positions: The first empty position is in row 1 and column 3, and the second in row 2 and column 3.

Step 3: Creating -connected bead positions
In this step, we followed the Definition 2 in creating bead positions on the nested chain abacus. Consider the 7-connected ominoes in Figure-

Step 4: Developing a connected partition of nested chain abacus
Using the 's obtained from step 3 we produce a partition called the connected partition which represents the nested chain abacus for -connected ominoes with columns and rows for . The transformation process of the 's into the connected partition is as follows: , , where Then ( ) is a connected position with e columns and r rows, where and ∑ . Consider Figure-2, where we illustrates the process of finding connected partition for 7-connected ominoes. Since , = then, . Based on the algorithm discussed earlier, we present a unique expression for n-connected ominoes, as shown in the Theorem 1. To prove it, we need Proposition 1 and Lemma 1. is a connected partition represented by the nested chain abacus with columns, rows and bead positions, such that . Meanwhile, is a connected partition representing the nested chain abacus with with columns, rows and of beads positions. Hence, is an associator with exactly one connected partition. Theorem 1: For any form of -connected ominoes, there exists a unique connected partition representing it with rows and columns . Proof. On the contrary, suppose that and are two nested chain abacus with columns, rows and beads representing one form of -connected ominoes. Based on the algorithm in step 2, there exist such that where , and . Since represents a -connected ominoes w.r.t minimal frame with columns and rows, so . Meanwhile, represents an -connected ominoes w.r.t minimal frame with columns and rows, then where is a location in the minimal frame containing an omino, and for . Based on Proposition 1, is injections so . Thus, any form of -connected ominoes is represented by exactly one . Based on Lemma 3, there exists a unique connected partition represented by , thus there exists a unique connected partition that represents -connected ominoes.
Previously, there was no representation for every shape of -connected ominoes. However, by using nested chain abacus, we can associate each of the four shapes of -connected ominoes with a connected partition using the nested chain abacus. The result of the representation is shown in Figure-3 where each shape is represents by a unique connected partition.

Topological Structure of Nested Chain Abacus
In this section, three design structures of the nested chain abacus are introduced; rectangular nested chain abacus, rectangle-path nested chain abacus, and square-singleton nested chain abacus. We begin by discussing the construction of rectangular nested chain abacus.

Rectangular nested chain abacus
The rectangular nested chain abacus consists of rectangular chains. Theorem 2 clarifies the construction of rectangular chains in nested chain abacus.

Theorem 2
Let there matrix that represents bead positions and empty bead positions in the nested chain abacus with columns, rows and chains. Then, 1. A vertical rectangular chain is an arrangement of the bead positions and empty bead positions in a vertical rectangular format in the nested chain abacus, such that the element chain is where is an even number and for . 2. A horizontal rectangular chain is an arrangement of the bead positions and empty bead positions in a horizontal rectangular format in the nested chain abacus, such that the elements chain is { } where is an even number and for .

Proof.
1. A vertical rectangular chain starting from the th column will start at the th row because rows 1 to will be covered by the vertical rectangular chains starting from rows 1, 2,...., . After the starting point, we will come down along the same column so the row numbers will be changing and will come down till the th row from the end, so that it will be the th row from the beginning. Now, in the th row, we go to the th column and then we should cover the chain by coming down till the th . So, two vertical columns have been covered. Now, from the starting point in the th row, we should cover up the chain on the right of it on that row till the th column; so that the row will remain fixed and the columns will vary till we reach th column in the th row and cover the chains in the th row by keeping the row fixed and varying the columns from to . This is basically how it is done: then ' to form the two vertical chains, and then to cover the rest; and ' ' . Thus the element chain will be: where is an even number and for . 2. Now, at the horizontal rectangular chain, the same situation arises so we cover the row chain starting from the th column and th row, then we cover up the chains in the ' th row and follow the same procedure to cover up the chains. This is basically how it is done: then to form the two horizontal chains, and then to cover the rest; ' and . and thus the element chain will be; where is an even number and for .

Rectangle-path nested chain abacus
The rectangle-path nested chain abacus consists of rectangular chains and a one path chain. Theorem 3 clarifies the construction of rectangle-path chain in nested chain abacus.

Theorem 3
Let the matrix A represents bead positions and empty bead positions in the nested chain abacus with columns and rows. Then, 1. A vertical-path chain is an arrangement of some bead positions and empty bead positions in column in the nested chain abacus, such that the elements in the chain are where , is odd and is a positive integer. 2. A horizontal-path chain is an arrangement of some bead positions and empty bead positions in the row in the nested chain abacus, such that the elements in the chain are where , is odd and is a positive integer. Proof. Each chain covers two rows and two columns, namely the th column and the th column, and for the vertical-path chain the th column and the th column become the same, and thus we get; or, , and similarly for the horizontal-path chain . So the vertical path chain will be in column and will start from th column and th row and here is so the rows start from the th row. and now it will continue till the th row. By putting the value of we get; = so the last row will be and thus we have the chain = where , is odd and is a positive integer. The same method is applied for the horizontal-path chain. From Theorem 3, there are two design structures of the rectangle-path nested chains abacus.The vertical rectangle-path nested chains with vertical rectangular chains and one vertical-path chain is a nested chain abacus with and is odd, where and are the number of the columns and rows, respectively. Example 3.3 provides the illustration of this design structure.

Crollary 1
Let be a nested chain abacus with column, rows and rectangular chains. Then is the arithmetic sequence for the number of positions in the nested chains abacus, with as the common difference of successive terms where is the number of positions in chain and for .
Proof. Let and represent the number of positions in chain and chain , respectively, where , where if the nested chain abacus is a vertical nested chain abacus and if the nested chain abacus is a horizontal nested chain abacus, as mentioned in Theorem 2. Thus .

Singleton nested chain abacus
The singleton nested chain abacus consists of rectangular chains and a singleton chain. Theorem 4 clarifies the construction of Singleton chain in nested chain abacus.

Theorem 4
Let the matrix A represents bead positions and empty bead positions in the nested chain abacus with columns and rows, where and is odd. Then, singleton chain is a position located in column and row .
Proof. Each chain covers two rows and two columns, the th column and the( th column, and for the singleton chain the th column and the ( th column become the same, and thus we obtain or, and similarly for the rows.

Conclusion
We have developed an algorithm for the construction of a nested chain abacus, as a representation of the previously described -connected ominoes [7,8,9]. At this stage, two questions can be asked: • Could we use the topological stracture of nested chain abacus to classify classes for -connected ominoes? • Could we use the design structure of nested chain abacus to construct the generating function for -connected ominoes?