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Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
DOI: 10.24996/ijs.2021.62.10.23
___________________________________
*Email:hanansalim73@uomosul.edu.iq
3648
Splitting the e-Abacus Diagram in the Partition Theory
Hanan Salim Mohammed*, Nadia Adnan Abdul-Razaq
Department of Mathematics, College for pure Sciences, Mosul University, Nineveh , Iraq
Received: 25/10/2020 Accepted: 31/3/2021
Abstract
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.
Keywords: Partition Theory, Abacus diagram, β-number, The beads.
e
Abacus diagram
and Mahmood Mahmood
e
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].
ISSN: 0067-2904
Mohammed and Abdul-Razaq. 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.
1
2
……
……
……
……
Mahmood [5] defined Abacus James diagram and - numbers 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.
2. 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 .
2.2. 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:
Table 1- Splitting e-Abacus diagram if into
Run1
Run2
Run3
Run4
Run1
Run2
Run1
Run2
0
1
2
3
0
1
0
1
4
5
6
7
2
3
2
3
8
9
10
11
4
5
4
5
12
13
14
15
6
7
6
7
.
.
.
.
.
.
.
.
`
.
.
.
.
.
.
.
.
Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
3650
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.
2.3 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.
Table 2- Representation of -numbers on the Abacus diagram
Run1
Run2
Run3
Run4
Run1
Run2
Run1
Run2
1
•
2
•
•
•
•
3
•
•
•
•
•
•
•
4
•
•
•
•
5
•
•
6
•
•
7
8
•
•
•
•
•
9
10
•
•
•
(10, 83, 52, 4, 34, 22, 1)
(5, 3, 22, 12)
(4, 32, 22, 13)
The partition of each split can be calculated in the case where e = 4, as shown in Table 3.
Table 3- Calculation of the partition of split (1) and split (2) in the case where e = 4
Row
Split(1)
Row
Split(1)
1
1
2
2
3
3
4
4
5
There are no beads
5
,where
6
6
2.4. 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.
Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
3651
Table 4- The general rule to find the partition of each split in the case where e = 4
Row
1
=1
2
Where
Where
3
Where
.
.
.
Where
.
.
.
The suggestion can be clarified when e = 4, as shown in Table 5.
Table 5- Clarification of the general formula when e = 4
Split(1) :
Split(2) :
1
if there is bead
if it is a vacuum
2
minus (positions of (1) row of split(2)
minus all the beads in 1 row of split (2))
minus (positions of 1,2 row of split 1 minus all
the beads in 1,2 row of split (1))
3
minus (positions of 1,2 row of split 2 minus
all the beads in 1,2 row of split (2))
.
.
It stops when the last void is reached before last
bead
.
.
.
.
It stops when the last void is reached before last
bead
2.5. 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.
Table 6- Splitting Abacus diagram if
Run1
Run2
Run3
Run4
Run5
Run1
Run2
Run1
Run2
Run3
0
1
2
3
4
0
1
0
1
2
5
6
7
8
9
2
3
3
4
5
10
11
12
13
14
4
5
6
7
8
15
16
17
18
19
6
7
9
10
11
20
.
.
21
.
.
22
.
.
23
.
.
24
.
.
8
.
.
9
.
.
.
.
.
.
.
.
The numbers, if as in the previous example, can be represented as
Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
3652
Table 7-Representation of The numbers on the e-Abacus diagram if
Run1
Run2
Run3
Run4
Run5
Run1
Run2
Run1
Run2
Run3
1
2
3
4
5
6
7
8
9
10
(5,3,2,1)
Table 8- The calculation of the partition of split (1) and split (2) in the case where e=5
Row
Split
Split
1
where
2
where
where
3
where
where
4
There are no beads
where
5
where
where
2.6. Splitting Abacus Diagram in the Case Where
If , then the Abacus diagram is split into ,
or and so on.
Now, we can select the splitting, which includes three splits, as shown in Table 9.
Table 9- Splitting Abacus diagram if
Run1
Run2
Run3
Run4
Run5
Run6
Run1
Run2
Run1
Run2
Run1
Run2
0
1
2
3
4
5
0
1
0
1
0
1
6
7
8
9
10
11
2
3
2
3
2
3
12
13
14
15
16
17
4
5
4
5
4
5
18
19
20
21
22
23
6
7
6
7
6
7
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
Table 10- Representation of the numbers on the Abacus diagram if
Run1
Run2
Run3
Run4
Run5
Run6
Run1
Run2
Run1
Run2
Run1
Run2
1
•
2
•
•
3
•
•
•
•
•
•
•
4
•
•
•
•
•
•
Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
3653
5
•
•
6
7
8
•
•
•
•
•
9
10
•
•
•
•
•
We can calculate the partition of each split, as shown in Table 11.
Table 11- Calculating the partition of split (1), split (2), and split (3) when
Row
split
Split
split
1
Where
Where
2
Where
Where
Where
3
Where
Where
There is no beads
4
Where
Where
Where ,4
2.7 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.
Table 12- The general rule for finding the partition of each split in the case where
Run
Split(1)
Split(2)
Split(3)
1
=1
=0 or 1
=0 or 1
2
3
4
.
.
.
.
.
.
.
.
.
Mohammed and Abdul-Razaq. Iraqi Journal of Science, 2021, Vol. 62, No. 10, pp: 3648-3655
3654
Note: If the splits are not equal, we apply the same previous rule.
2.8. The General Rule for Finding Partitions for and
By 2.2-2.7, the proof of the general rule of splitting a large e-Abacus diagram is defined by
Table 13 f.
Table 13-The general rule for finding the partition of each split in the case of large
Abacus diagrams.
Run
1
.
.
.
.
2
…
3
.
.
.
.
.
.
.
.
…
.
.
.
…
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.
Acknowledgement
We express our great thanks and gratitude to the chancellorship of Mosul University,
represented by the Deanship of the College of Education for Pure Sciences, Department of
Mathematics for supporting this work.
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Math. Soc., Wellington, 1979.
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3655
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