A Tile with Nested Chain Abacus

This study had succeeded in producing a new graphical representation of James abacus called nested chain abacus. Nested chain abacus provides a unique mathematical expression to encode each tile (image) using a partition theory where each form or shape of tile will be associated with exactly one partition.Furthermore, an algorithm of nested chain abacus movement will be constructed, which can be applied in tiling theory.


Introduction:
About 15 years after James abacus was introducedJames abacus diagram is a graphical representation of a special type of non-increasing sequenceµ = (µ 1 , µ 2 ,...,µ b )called the partition of twhere b is the number of partition parts and t is any positive integer. The James abacus diagram configuration for beta numbers β 1 , β 2 , β 3 ,...,β b can be created by rearranging these numbers on the runners, where β i = µ i + b − i for1 ≤ ≤ ) 1 .A new abacus was established by adding one empty column to the James abacus called ( + 1)-abacus. It was proven in a theorem related to the determination of the decomposition numbers for any where is a prime positive integer greater than or equal to 2 2 . James and Mathas investigated several relationships between the original and the new partition. Fayers (2007) contracted a similar theorem but with different weight of partition by adding 'full' column to the James abacus 3 .
The parallelism idea of the previous theorem in 3 was used by considering a finite composition of weight of the partition 4,5 . Fayers, expands the use of the James abacus to compute the -regularisation of a partition by implementing another movement. This movement is considered to be more complicated than the previous movement. A bead moving from to − and a bead from − to , k = 1, … , , where positions − and are empty beads while positions − and are beads as long as the volume 1, … , where cis the number of themovement of the bead positions of James abacus, such as a work by Wildon (2008)who discovered a new way to find the conjugate of any partition by reflecting James abacus in its leading diagonal 6 .
The literature on James abacus shows the constraction of a variety of diagrama beads on movement of the bead positions of James abacus, such as a work by 7 who discovered a new way to find the conjugate of any partition by reflecting James abacus in its leading diagonal.
Scan movement was the development in the case the bead jumps positions to the right, which is used to prove the combinatorial definition of Schur polynomials equivalent the algebraic definition of Schur polynomials 8 . Another movement of the bead was constracted to give a combinatorial proof of a plethstic generalization of the Murnaghanâ Nakayama rule called single-step bead move. In this movement, all bead positions will be changing locations from position to position − above it in the same column such that − ⩽ 0 and − is empty bead position 9 .
Tingley, places all beads on the horizontal axis and then moves one bead exactly one step to the right in the corresponding row of beads, possibly jumping over other beads. Then, he put the beads into groups and rotating each group 90 degrees anticlockwise 10 .
Meanwhile, a new diagram was found by applying Brauer algebra on James abacus along with a consideration of the Temperley-Lieb algebra 11 . In addition a new abacus called neated chain abacus was consract. The construction of the new diagram is developed by first converting the original James abacus diagram to matrix form. James abacus diagram positions are divided into several nested chains 12,13,14 . Many movement were construct to converts one tile (image) into another or converts one tiling of a lattice region into another 15,16 .
In this work, a new abacus depending on James' abacus idea to represent any tile as a discrete object was constructed. In addition, three types of chain movement are developed. These movements have been used in tiling.

Definitions And Terminologies
This section briefly discusses some basic steps, definitions and theorems of tile using nested chain.The graphical form of tile with respect to a minimal frame, enables us to define object positions and empty object positions in terms of rows and columns of the minimal frame. Then the object positions and empty object positions as bead positions and empty bead positions respectively were redefined, which enables us to apply the concept of beads on an abacus to represent any tile on a nested chain abacus. Definition 1(15): A tile is a plane geometric figure formed by one or more ominoes. Definition 2(17): A minimal frame is a minimal rectangle containing the tile itself, s.t. each column and row contains at least one omino where < for n ominoes, r rows and e columns. Definition3(17): Let tile in a minimal frame have e columns and r rows. The function ( , ): × → such that, if location (m, j) in the minimal frame contains an ominoes, then = ( , ) = ( − 1) + ( − 1) for = 1,2, … , where e and r refer to the number of the rows and columns of the minimal frame respectively, for 1 ⩽ ⩽ and 1 ⩽ ⩽ . Definition 4 12 : A nested chain abacus is an abacus with columns, rows and of bead positions which satisfy the following conditions: 1. The columns are labelled from left to right as 0,1, … , − 1. 4. Each column and row has at least one bead position. Table 1 corresponding placement of position numbers on the nested chain abacus with columns number from 1 to − 1 and rows number from 1 to − 1. Nested Chain Abacus 17 Following anew representation of n-connected ominoes called nested chain abacus construction Step-I: Establishing a graphical form of tile w.t.r. minimal frame. 1-The abacus with columns and r rows is considered a tile (picture) with the minimal frame (rectangular forms), and the bead as well as empty bead positions which are considered squares in different colors where the bead positions constitute the image of the picture and the empty bead positions constitute the background. 2-Identify the first column (leftmost row), last column (rightmost row), first row (topmost column) and last row (bottommost column) with at least one ominoe as a minimal frame. The columnsnumbered from the leftmost, worked from left to right 1,2,…, and numbered the rows from topmost to bottommost 1,2, … . , in the minimal frame where r and e number of rows and columns respectively.
Step-II: Mathematical expression for tile.
In this step, a direction to obtain a nested chain abacus with respect to the minimal frame is created.Identify the first ominoe located in the topleftmost, from left to right, working down from the top-leftmost ominoe to the bottom in the minimal frame.
Step-III: Connected partition of nested chain abacus. following Definition 3 bead positions on the nested chain abacus has been disputed. According to Step II, beginning at the top-leftmost ominoe of the minimal frame and the rest of the ominoes.
Step IV: Constructing a connected partition of the nested chain abacus Using the 's obtained from Step 3, aconnected partitionwhich represents the nested chain abacus with beads, columns and r rows for = 1,2, … , called connected partition is produced.

Figure1.Modeling the 4 shapes of family of Tetrominos using nested chain abacus
The nested chain is considered as tile (picture) with rectangular forms and the empty bead as well as bead positions are considered squares in two colors where the bead position constitutes the tile (image) in the two colors where the bead position constitutes the image of the tile ( image) and the empty bead position constitutes the background.

Nested chain movement
This proposed method is called nested chain movement. That is, any bead position as well as empty position can be used as an initial bead to be moved from one location to another location and the rest of the beads will follow to form a chain. The number of chains in a diagram depends on column e and row r. In new movement any bead position can be bypass one location or more. The proposer movement is used to convert any tiling into any other by means of these movement. Definition 5: Chain movement (Ch) is a moving movement when ∈ ℤ, which is in anticlockwise direction for all bead positions in chain in the nested chain abacus with columns and rows where0 ⩽ ⩽ − 1and 0 ⩽ ⩽ − 1.
where < , is odd and is a positive integer.
• Horizontal-path chain is an arrangement of bead and empty bead positions in row where < , is odd, and is a positive integer.

Remark 7:
In the chain movement the beads will move by a specific distance. A bead located in the chain i and in the column i will move to the downwards if c is a positive integer. While, a bead located in the row r-i+1 will move to the rightwards if d is a positive integer. Meanwhile, the bead located in the column e-i+1 will move to the upwards if c is a negative integer. Finally, the bead located in row i will move to the leftwards if d is a positive integer.  where is a positive integer. Then, the position will move downward.
where is a positive integer. Then, the position will move rightwards.
where is a negative integer. Then the position will move upwards.
where is a negative integer. Then the position will move leftwards.
In our movement a position in the chain as the initial point was selected. When this point is moved rotationally anticlockwise to a new location, all other positions in the rectangle chain will move rotationally to a new location accordingly. Definition 10: Nested chain abacus movement is a chain movement in one or more chains in the nested chain abacus.
Next the movement of the bead positions inside the chains is construct. Lemma 9 provides the basic concept for bead position movements.

movement in Chains
There are three cases of movements based on three types of chains: movement in rectangle chain, movement in path chain and movement in singleton chain. Next, a new movements in nested chain abacus is constructed.
Corollary 18Let ( ) be an element in the horizontal rectangle-path chain in the nested chain abacus with e columns, r rows and c chains represented by matrix × . Then, ℎ: where 1 ≤ ≤ − + 1. Proof : it follows immediately from Lemma 16 and theorem 17 In the following theorem the number of possible movements in the horizontal-path chain were determined. Theorem 19: Let a m j be an element in the horizontal-path chain in the nested chain abacus with e columns, r rows and c chains represented by matrix × . Then, the total number of movements of each position in the path-chain is − + 1 Proof. Let a m j be a position in the horizontal-path for nested chain abacus with e columns, r rows and c chains. Based on Lemma 44, a m j can be moved rightwards and leftwards depending on m. Therefore : Let a m j be an element in the horizontal-path chain in the nested chain abacus with e columns, r rows and c chains represented by matrix A r e . Then, the total number of movement of each position in the path-chain is − + 1. Proof. Let a m j be a position in the horizontal-path for nested chain abacus with e columns, r rows and c chains. Based on Lemma 44, a m j can be moved rightwards and leftwards depending on m. Therefore Thus a m j will skip 2 − + 1 2 − + 1 2 + 1 = − + 1

Nested Chain Abacus Movement Algorithm
This section begins by formulating the conceptual framework used to structure a new abacus. This is done by converting nested chain abacus to matrix form where any bead position as well as empty bead position in the nested chain abacus is represented as the element of a matrix and then applying the new movement. This is followed by the development of three different types of nested chain abacus movement which are single nested chain abacus movement with = 2 (SNC2-Movement), stratum nested chain abacus movement with > 2 (SNC-Movement) and multiple chain movement (MNC-Movement).

SNC2-Movement
A movement of a nested chain abacus with a chain is called SNC2-Movement construction.
Step 1: Convert the nested chain abacus with beads and one chain into ×2 .
Step 2: Select as an initial point where is an element in the × matrix.

SNC-Movement
A movement of a nested chain abacus with chains is called SNC-Movement construction. In SNC-Movement, the chain movementhas been employed in only one chain.
Step 1: Convert the nested chain abacus into × .
Step 2: Select the chain and then as an initial point where ⩽ ⩽ − + 1 and ⩽ ⩽ − + 1.  Fig. 3. Consider Fig. 3 of 19 beads the new location of beads is as shown in Table 3.

MNC-Movement
A movement of a nested chain abacus with chains called MNC-Movement construction.
Step 1: Converted the nested chain abacus into × .

Conclusions:
In this research a new combinatorial interpretation called nested chain abacus was used to encode any tile. Furthermore, the research presented and formated three type of movement. Based on theses movementsan Algorithems of SNC2-Movement, SNC2-Movement and MNC-Movementwere developed to constract new abacus (tile).