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Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375 DOI: 10.24996/ijs.2021.62.7.26
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* fatima.esp87@student.uomosul.edu.iq 2369
Replacing the Content in -Abacus Diagram II
Fatmah Ahmed Basher*1, Ammar Seddiq Mahmood2
Department of Mathematics, College of Education for Pure Science, University of Mosul, Mosul-Iraq
Received: 12/8/2020 Accepted: 23/1/2021
Abstract
In our normal life, we sometimes need a process of replacing something with
another to get out of the stereotype. From this point of view, Mahmood’s attempted
in the year 2020 to replace the content in the first main e-abacus diagram. He found
the general rule for finding the value of the new partition after the replacement from
the original partition. Here we raise the question: Can we find the appropriate
mechanisms for the remainder of the main e-abacus diagram?
Keywords: Partition theory, Composition, Partition, -abacus diagram
e
- II
e
1. Introduction
Since the emergence of the topic of e-abacus diagram by James [1], many researchers have been
studying many of the traits that exist in the first place, as well as studying some changes that can be
used in many areas. Fayers [2-3], Mathas [4], and others presented many relationships that made this
topic the focus of much interest. Here we study the possibility of the replacement of content within
this diagram and knowing the general behavior and its effects on the mathematical concepts of this
topic.
Let be a nonnegative integer. “The composition where
is the number of times that appeared, z=1, 2, …, m of which is the sequence of non-negative
integers such that
.” The composition is called a partition of if
is fixed as a partition of and we define that, . The set
is said to be the set of - number for [4]. Let e be a positive integer number greater
than or equal to 2, then we can represent numbers by a diagram called e-abacus diagram[1], as shown
in table 1.
ISSN: 0067-2904
Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375
2370
Table 1- e-abacus diagram
Runner 1
Runner 2
Runner e
0
1
e-1
e
e+1
2e-1
2e
2e+1
3e-1
where every will be represented by a star and the rest of the sites are denoted by (-). In fact,
the definition of e-abacus diagram will lead us to the fact of the presence of an infinite number of
diagrams that are all suitable for any partition according to the value of e. For example, if
and if we chose e=3, then we have many of e-abacus diagrams of this
partition, as follows:
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
To develop a specific method to control the number of these diagrams, Mohammed [5] provided the
following: “For any partition of r with n parts, let where , is said to be
guides for this partition”. For example, let ,1), then the diagrams in tables 2 and 3 hold:
Table 2- 2-abacus diagrams for ,1)
0
1
-
-
2
3
-
-
-
-
-
4
5
-
-
-
-
-
-
6
7
-
-
-
-
-
8
9
-
-
-
-
10
11
-
-
-
-
14
15
--
-
-
-
16
17
-
-
-
-
18
19
-
-
-
-
-
20
21
-
-
-
-
-
-
Table 3-3-abacus diagrams for ,1)
0
1
2
-
-
-
-
3
4
5
-
-
-
-
-
-
-
-
-
6
7
8
-
-
-
-
-
-
-
9
10
11
-
-
-
-
-
-
12
13
14
-
-
-
-
-
-
-
15
16
17
-
-
-
-
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-
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-
Any e-abacus diagram for each guide is said to be main or guide e-abacus diagram. Then there
exist e of these diagrams; see a previous article [6] for more information about the technology that
these diagrams take to become in this order.
Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375
2371
Mahmood [7] submitted an idea to replace the content in e-abacus diagram, denoted by ; Table 5
shows this idea for the above example.
Table 4- -abacus diagram
e = 2
e = 3
e = 5
-
-
-
-
-
-
-
-
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-
-
-
,1)
,1)
,1)
Through this, he was able to know the general rule for finding the value of partition after the
proposed replace process, but only for the case of (or for the first main e-abacus diagram). In this
paper, we will replace the content and find the value of partition for any value greater than or equal to
.
2. Explanation of the phenomenon
The location of in the row k and column l where (will play a fundamental role in
the end position of the original diagram. We will then replace it considering what it is there.
Location of ( = +((no. of parts of -1) (1)
Which is exactly equal to
(Note that, any part of power 0 or 0 of power 1 is not appearing here). (3)
Case I: In this case, we will use all the relationships listed by Mahmood in [6] and [7]:
Rule (2.1): The partition after replacing the content in1st e-abacus diagram will be
.
Proof: Through (1) and (2) in above, this location in the original diagram will be a ( and, after the
replacement, it will turn into a (-). Then we have three possibilities that depend entirely on e-l: “The
first possibility is that, if a ( is found after it, then it is calculated, unless this is in the condition (3)
that makes e-l equal to zero, which is the second possibility. Otherwise, it indicates the existence of (-
) only and this is the last possibility, which has no effect on mathematical relationships.”
Rule (2.2): The partition after the replacement of the content in 2nd e-abacus diagram will be
.
Proof: By using all the remarks in [6], we notice that one is always added to the top of the first
column, as well as a (- ) to the bottom in the case of e=2, two ( - ) in the case of e=3, …, and so on
downwards until adding t-1of the ( - ) when e=t in the second main diagram. After the replacement, all
of this will change to the opposite, which explains the existence of the number 1 that is always added
to the relation and likewise in the first term of power (e-l) +(e-1)
Rule (2.3): The partition after the replacement of the content in the 3rd e-abacus diagram will be
.
Proof: This is similar to the Rule (2.2), except that we always in front add two ) from the top in the
first column and we add from the bottom two ( - ) if e=3, four ( - ) if e=4, …, ((2t-4) of ( - )) when
e=t)
For example, if , if we chose e=4, then we have four main e-abacus diagrams, as
follows:
Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375
2372
1st
2nd
3rd
4th
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-
of 1st
of 2nd
of 3rd
of 4th
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-
Then, by using the same prove of Rules (2.2) and (2.3), we have the following:
Rule (2.4): The partition after the replacement of the content for any X e-abacus diagram, it will be
Case II: According to [6], there is a certain mechanism to find the main e-abacus diagram, where we
notice that in the end there is always the presence of a (-) which does not affect the value of partition
at all, but it will make a large change in the case of substitution of the content as these will be in the
form of beads. So, we have two cases. The first case is that we keep (-) as it is. The second case is to
delete it basically before replacing. It might be natural that there is a question about case I, which is “if
there exist (- ) after the last
in a given diagram, why do we reposition it in the next diagram since it will not basically affect the
main partition?”. This question obliged us to study this case and, before going into its details, we will
give an example that will clarify what we mean:
Case
I
1st
2nd
3rd
4th
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Case
II
1st
2nd
3rd
4th
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-
Through the above, we note that the difference between the two cases is that, in Case I, we have ( -
) at the end of the all main diagrams, which will have a significant impact in calculating the general
Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375
2373
rule after the replacement. In Case II, it will behave differently to the first case! Consequently, we
must study the general behavior in every main diagram as follows:
i- Let be the number of columns in main e-abacus diagram where .
ii- Location of Location of ( = +((no. of parts of -1) = (k – 1)e + (.
iii- Where h = 1, 2, …, e – 1, then
.
Now, we obtained the following rules by using (i-iii) in above and (2.1-2.3), unless the last rows are
with (-):
Rule (2.5): The partition after the replacement of the content in the 1st e-abacus diagram will be
.
Rule (2.6): The partition after the replacement of the content in the 2nd e-abacus diagram will be
.
Rule (2.7): The partition after the replacement of the content in the 3rd e-abacus diagram will be
,
where
In general, we have the following rule:
Rule (2.8): The partition after the replacement the content in any (X+1), the e-abacus diagram will be
,
where
Then, without using any diagrams, we have the applications of rules of an example arbitrary, as
presented in Tables- 5 and 6.
Table 5- The rules of Case I and Case II for where e=3
Case I
Case II
1st
2nd
3rd
1st
2nd
3rd
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Basher and Mahmood Iraqi Journal of Science, 2021, Vol. 62, No. 7, pp: 2369-2375
2374
Table 6- The rules of Case I and Case II for where e=4
)
Case I
Case II
1st
2nd
3rd
4th
1st
2nd
3rd
4th
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
3. Conclusions
This paper has reached several conclusions. First, each diagram in gives a new different partition
in the main e-abacus diagram.
Second, it is possible to have a similarity in the value of partition in only for the case 1st e-abacus
diagram.
Third, for the first time, the content replacement method is used and the insights of this method are
carefully studied,. This will provide later the possibility of adopting it as a type of encoding or
encryption in many applications on the topic of partition, thus opening new horizons for scientific
research in this direction. See [8-15].
Acknowledgment
We thank the University of Mosul/College of Education for Pure Science for their moral support
during the preparation of this research.
References
1. James, G. D. 1978. Some combinatorial results involving Young diagrams. Mathematical
proceedings of the Cambridge Philosophical Society, 83: 1-10. http://doi.org/10.1017 /S0305
004100054220
2. Fayers, M. 2007. Another runner removal theorem for r-decomposition numbers of Iwahori-Hecke
algebra and q-Schur algebra, Journal of Algebra, 310: 396-404.
3. Fayers, M. 2009. General runner removal and the Mullineux map, Journal of Algebra, 322: 4331-
4367.
4. Mathas, A. 1999. Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group.
University Lecture Series, 15, http://doi.org/10.1090/ulect/015/02
5. Mohammed, H. S. 2008. Algorithms of the Core of Algebraic Young’s Tableaux, M. Sc. Thesis,
University of Mosul (Iraq).
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6. Mahmood, A. S. 2011. On the intersection of Young’s diagrams core, Journal of Education and
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1-6. http://doi.org/10.4236/oalib.1106211
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Research in Science, Technology Engineering, 6(12): 53-62. http://www.erpublications. Com /
our -journals-dtl.php/pid=1
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