Splitting the e-Abacus Diagram in the Partition Theory

In the partition theory, there is more then one form of representation of dedication, most notably the Abacus diagram, which gives an accurate and specific description. In the year 2019, Mahmood and Mahmood presented the idea of merging more than two plans, and then the following question was raised: Is the process of separating any somewhat large diagram into smaller schemes possible? The general formula to split e-abacus diagram into two or more equal or unequal parts was achieved in this study now.


Introduction
Despite all studies and researches that have been presented on the subject of partition theory, the subject of large partition diagrams makes it difficult to conduct studies about coding letters and other topics, in addition to the time and great effort made to fulfil that. Therefore, a method was employed that assists the study of these large diagrams, related to partition theory, after splitting the diagram into a number of small diagrams, determining the corresponding diagrams for each new case. According to specific laws and rules and depending on the value of , we can split e into , and it will be quite natural that these parts will be equal or unequal if the value of chosen is an even number. On the other hand, if value is odd, then the resulting splits will be unequal. All the processes of splitting are based on the rules and laws proved in this research.
Let n be an integer positive number. A sequence of non-negative integers, such that [ ] ∑ , for ≥1 and , then the sequence is called a partition [1]. The partitions theory was discovered by Andrew [2,3].
Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655 3649 The partition theory is considered as the cornerstone of algebra. James [4] defined , which are called beta numbers, for and as a non-negative integer. The Abacus has vertical runners, labelled as from left to right from top to bottom. Mahmood [5] defined Abacus James diagram andnumbers as the values of , which are called the "guides", where is the number of the parts of the partition of , and the guide represents the "Main diagram" or "Guide diagram". All additions in this area motivated the researchers to create several ideas in the partition theory, including the direct application of this topic. One of the most vital applications are what Ilango and Marudh discussed [6], in which they depended on the idea of the nodes and voids when using the sonar to obtain the best images and then treating the patient as quickly as possible. Also, Andrews [7] applied his idea on the Tile domino surfaces, which he regarded as the most important application in the composition of Aztec diamonds. The subject of the main diagram that has been expanded by Mahmood and others [8] presented new additions that resulted in more important topics. Also, new additions to the topic of partition theory and β-numbers [9] led to the emergence of the idea of coding the Syriac letters in 2017 [10,11]. In 2018 and 2019, both Mahmood and Mahmood [12,13] presented the idea of coding English letters and, where adding these letters according to the rule was remarkably useful. In fact, the idea of coding is common for many researches and other different topics [14]. As for the current topic, i.e. the partition theory, which has been expanded widely, it has become difficult to study large -Abacus diagrams and -numbers [15,16]. Hence, it was decided to split large -Abacus diagrams into several smaller tables and then to find out the corresponding partition for each split to facilitate future studies. It is quite natural that the process of splitting is not merely the division into two or more equal or different parts, because if we choose e as an odd number, then we cannot divide the diagram equally at all, which is what we will present in this research.

The Proposed Method 2.1 Splitting e-Abacus
There are large Abacus diagrams with big partitions, which are hard to study or to perform mathematical operations on them. Thus, it is better to split these Abacus diagrams such that each part may represent a partition and can be easily read from right to left. The basic rule of this partition is . The reason is that the smallest that can be chosen is , and therefore, any diagram to be divided must be at least .

Splitting e-Abacus in Case e=4
If , Abacus diagram should be split into , as in the Table 1, where the partition can be read according to the following: Let be the space that precedes the node in the original partition, which consists of four columns. Assume that is the number of beads in the line for all columns, except for the column itself. An example for that is the following.

Example
If is a partition, the -number of this partition where such that are ), and then we can represent it on the Abacus diagram in the form of a bead, as mentioned by James [4]. The following Table 2 shows this case. The partition of each split can be calculated in the case where e = 4, as shown in Table 3.

General Method for Splitting e-Abacus Diagram Where
The general rule for finding the partition is drawing the Abacus diagram, as shown in Table 4. . .

∑ ∑
The suggestion can be clarified when e = 4, as shown in Table 5.

Splitting e-Abacus Diagrams in Case
If , then Abacus diagram is split into , or the opposite, that we get two partitions, which can be represented on the Abacus diagram, as shown in Table 6.   and so on. Now, we can select the splitting, which includes three splits, as shown in Table 9. The β-numbers for the previous example, when e = 6, can be represented as (23,20,19,18,14,13,11,9,8,7,6,4,3,1), as shown in Table 10. We can calculate the partition of each split, as shown in Table 11.

A Suggested Method for Finding Partitions for , and
The general rule for finding the partition is drawing the Abacus diagram, where , as shown in Table 12.

Conclusions
Through the work performed in this research, a conclusion was reached in terms of finding the general formula to find the partition for each e-Abacus diagram, which results from splitting the large diagrams of any partition.