On (gαφ), - Regularisation of English Letters Partition

Abstract

Any partition it has two characteristics, the first is called regular and the second is regularisation, both of which were introduced by James in 1976. Through this research we will try to present a new type that greatly serves the proposed model of English letters that were presented by Mahmood and Mahmood in 2019.

Full text

Journal of Physics: Conference Series
PAPER • OPEN ACCESS
On (
gαφ
, - Regularisation of English Letters Partition
To cite this article: Alaa I. Alabaaweee and Ammar S. Mahmood 2021 J. Phys.: Conf. Ser. 1804 012031
View the article online for updates and enhancements.
This content was downloaded from IP address 45.134.241.5 on 02/03/2021 at 13:56

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
1
On 𝒈𝜶,𝝋-Regularisation of English Letters Partition
Alaa I. Alabaaweee1, Ammar S. Mahmood1.
1Department of Mathematics, College of Education for Pure Science, University of
Mosul, Mosul-Iraq.
Alaa.esp85@student.uomosul.edu.iq,
asmahmood65@uomosul.edu.iq
Abstract. Any partition it has two characteristics, the first is called regular and the second is
regularisation, both of which were introduced by James in 1976. Through this research, we will
try to present a new type that greatly serves the proposed model of English letters that was
presented by Mahmood and Mahmood in 2019.
Keywords: Partition, Young diagram, 𝑝 regular, regularization, e-Abacus Diagram.
1. Introduction
For decades, Partition theory has been an important source for many topics, particularly algebra.
Through this research, we will try to review the most important stations for which we seek to reach the
goal of achieving a regularisation of some concepts. Let 𝑟 be a non-negative integer. The partition
𝜔𝜔,𝜔,…,𝜔𝜃,𝜃,…,𝜃
(where 𝛾 the number of times 𝜃 appeared, z=1, 2, …, m) of 𝑟 is a sequence of non-negative
integers such that 𝜔𝜔⋯𝜔 𝑜𝑟 𝜃𝜃⋯𝜃 𝑎𝑛𝑑 |𝜔|∑𝜔

  ∑𝜃

 
𝑟, [1]. 𝜔 is often identified with its Young diagram, which is the subset 𝜔𝑖,𝑗|1𝑗𝜔 & 𝑖
1 of ℕ. The element of 𝜔 are called the nodes of 𝜔, we draw the Young diagram as an array of
boxes. For example, the Young diagram of 𝜔3, 3, 3, 2, 2, 1, 1 ,1, 3,2
,1
 is:
[𝜔
Figure 1. Young Diagram
There are many definitions of e-regular, which we mention
:
“A partition is said to be e-regular, if no e rows; of Young diagram, of it have the same length, [2].
We say that 𝜔 is e-regular if there is no 𝑖1 such that 𝜔𝜔0 and otherwise we say that 𝜔

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
2
is e-singular, [3]. A partition 𝜔 = 𝜃,𝜃,…,𝜃 is called e-regular if 1𝛾𝑒1, ∀ z=1, 2,
…, m, [4]”.
In this paper, we use the following: “For any partition 𝜔, let 𝜔 be the partition which is
conjugate to 𝜔; that is, 𝜔𝜔,𝜔,⋯ where 𝜔 is equal to the number of nodes in column c of
[𝜔. From the above example, [𝜔8, 5,3. A partition 𝜔 is e-restricted if 𝜔𝜔𝑒,∀𝑗1.
A partition 𝜔 is e-regular if its conjugate is e-restricted, [1]”; from the above example, 𝜔
3, 3,3, 2, 2, 1,1 ,1,  is 4-regular.
2. Regularisation
The notion of regularisation was introduced by [2]; for 𝑙1, we define the lth ladder in ℕ to be the
set of nodes 𝑖,𝑗 such that 𝑖𝑒1𝑗1𝑙. The regularisation of 𝜔 is defined by moving all
the nodes of 𝜔 in each ladder as high as they will go within that ladder, and always e-regularisation is
a partition. For example, if 𝜔5,4,2
,1
 then 2-regularisation=10,6,5,4,2 and 3-
regularisation=6,5, 4,3
,1
 as follows:
𝜔5,4,2
,1
 → 2-regularisation of 𝜔
1 2 3 4 5
1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 2 3 4 5 6 7
3 4 5 6 7 3 4 5 6 7
4 5 6 7 4 5 6 7
5 6 5 6
6 7
7
8
9
10
Figure 2. 2-regularisation of 𝟓𝟑,𝟒,𝟐𝟐,𝟏𝟒.
𝜔5,4,2
,1
 → 3-regularisation of 𝜔
1 3 5 7 9 1 3 5 7 9 11
2 4 6 8 10 2 4 6 8 10
3 5 7 9 11 3 5 7 9
4 6 8 10 4 6 8 10
5 7 5 7 9
6 8 6 8 10
7 7
8 8
9
10
Figure 3. 3-regularisation of 𝟓𝟑,𝟒,𝟐𝟐,𝟏𝟒.
James in [ ] proposed that a prime number p which will be in direct contact with p-regular. One of
the problems facing regularisation is that if p greater than the number of repeated rows in a Young
diagram then we cannot make any movement within a regularisation. An example of this case:
𝜔5,4,2
,1
 → 5-regularisation of 𝜔
1 5 9 13 17 1 5 9 13 17
2 6 10 14 18 2 6 10 14 18
3 7 11 15 19 3 7 11 15 19
4 8 12 16 4 8 12 16
5 9 5 9
6 10 6 10
7 7
8 8
9 9
10 10

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
3
Figure 4. 5-regularisation of 5,4,2
,1

The aim was to find a way to solve this dilemma, which we will call regularization for special type
, and this is what we will explain in the next section.
3. ( 𝒈𝜶,𝝋-Regularisation
In this part we will try to find a mechanism that links the primitive p in the concept of regularisation
with the new method, as follows: If 𝜑 is maximal value of 𝛾, then we call 𝒈𝜶,𝝋-regularisation if it
achieved the following equation:
𝑔𝑝𝜑𝛼
2 𝑖𝑓 𝛼0,1, ⋯,𝜑 𝑎𝑛𝑑 𝑔𝜑,
𝜑 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒;𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 𝑖𝑓𝜑 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 .
We will try to apply this technique to the proposed model on the use of the concept of partition on
the English letters by Mahmood and Mahmood in [ 5-6 ]; 𝑖𝑔1𝑗1𝑙, as follows:
Table 1. The Partition of Each English Letters
Letter partition Letter partition
A 11, 8,5
,2,1
 B 11,19,8,6
,5,3,1

C 13,12,9,5
,2,1
 D 12, 11,9,8,6,5,3, 1
E 12,8,6
,2,1
 F 12,8, 6,2,1

G 11,10,7
,6,2,1
 H 13,11,10,8, 7, 6,4,3,1
I 15, 12,8,4, 1 J 14,11,10,8,4, 1
K 15,13,11,10, 7, 5,4,3,1 L 17, 13,9,5,1
M 12, 9,8,7
,6,5
, 4,3,2,1 N 11,9, 8,7,6
,5,4
,3,1
O 12,11,8
,5
,2,1
 P 11,8,6
,5,3,1

Q 11,10
,8
,5
,2,1
 R 13,11,10,8, 6,5,3,1

S 13,12,7
,2,1
 T 14,10,6,2, 1
U 14, 12,10,9,7,6,4,3,1 V 16,13,12,11, 8,5
W 14,13,12,11, 10,9,8
,5 X 13,10,9,8,5,2,1
Y 16,12, 9,8,5 Z 13, 10,7,4, 1
In the following table, We will give 𝜑,𝑝 and 𝑔 for each letter, (No) means it is impossible for this
matter to happen ; 𝑔 is not integer number, in this case and (?) It means 𝑔 is not prime:
Table 2. 𝜑,𝑝 and 𝑔 for Each English letters
𝒈𝜶𝟕 𝒈𝟕
𝒈𝟔
𝒈𝟓
𝒈𝟒
𝒈𝟑
𝒈𝟐
𝒈𝟏
𝒈𝟎
𝒑
𝝋
Yes No
5
No
?
No
3
No
2
11
7
A
No
No
?
No
3
No
2
No
?
5
3
B
No
No
?
No
3
No
2
No
?
5
3
C
No
No
?
No
3
No
2
No
?
5
3
D
No
?
No
3
No
2
No
No
No
5
4
E
No
?
No
3
No
2
No
No
No
5
4
F
No
?
No
3
No
2
No
No
No
5
4
G
No
?
No
3
No
2
No
No
No
5
4
H
No
No
?
No
3
No
2
No
?
5
3
I
No
No
?
No
3
No
2
No
?
5
3
J
No
?
No
3
No
2
No
No
No
3
2
K
No
?
No
3
No
2
No
No
No
5
4
L
No
?
No
3
No
2
No
No
No
3
2
M
No
?
No
3
No
2
No
No
No
5
4
N
No
No
?
No
3
No
2
No
?
5
3
O
No
No
?
No
3
No
2
No
?
5
3
P
No
?
No
3
No
2
No
No
No
5
4
Q
No
No
?
No
3
No
2
No
?
5
3
R
No
No
?
No
3
No
2
No
?
5
3
S

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
4
No
No
?
No
3
No
2
No
?
7
5
T
No
?
No
3
No
2
No
No
No
3
2
U
No
?
No
3
No
2
No
No
No
3
2
V
No
?
No
3
No
2
No
No
No
3
2
W
No
?
No
3
No
2
No
No
No
2
1
X
No
No
?
No
3
No
2
No
?
5
3
Y
No
No
?
No
3
No
2
No
?
7
5
Z
3.846% 0% 3.846
%
50% .1546
%
50% 50% 0% 3.84
6%
percentag
e
Note (3.1):
It is quite evident that the value of 𝜑 will play the largest role in the possibility of
applying this kind of regularisation to each proposed partition. That is means, in some cases of 𝑔
we
will not have any possibility of a regularisation, as for the fact that the 𝑔 is not integer or not prime.
From the foregoing, each percentage of the possibility of applying a state of 𝑔, whether large or small,
has its distinct value in making a difference to this topic.
A=𝟏𝟏,𝟖𝟐,𝟓𝟕,𝟐,𝟏𝟑 → 𝒈𝟎 of A=𝟏𝟒,𝟏𝟑,𝟏𝟏,𝟗,𝟖,𝟔,𝟒,𝟐
1 2 3 4 5 6 7 8 9 1
0
1
1
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
1
4
2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
1
4
3 4 5 6 7 8 9 1
0
3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
4 5 6 7 8 4 5 6 7 8 9 1
0
1
1
1
2
5 6 7 8 9 5 6 7 8 9 1
0
1
1
1
2
6 7 8 9 1
0
6 7 8 9 1
0
1
1
7 8 9 1
0
1
1
7 8 9 1
0
8 9 1
0
1
1
1
2
8 9
9 1
0
1
1
1
2
1
3
1
0
1
1
1
2
1
3
1
4
1
1
1
2
1
2
1
3
1
4
Figure 5. 𝑔 of A
A=𝟏𝟏,𝟖𝟐,𝟓𝟕,𝟐,𝟏𝟑 → 𝒈𝟐 of A=𝟏𝟏,𝟗,𝟖,𝟕𝟐,𝟔,𝟓,𝟒,𝟑𝟐,𝟐,𝟏𝟐
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 18
3 5 7 9 11 13 15 17 3 5 7 9 11 13 15 17
4 6 8 10 12 4 6 8 10 12 14 16
5 7 9 11 13 5 7 9 11 13 15 17
6 8 10 12 14 6 8 10 12 14 16
7 9 11 13 15 7 9 11 13 15
8 10 12 14 16 8 10 12 14
9 11 13 15 17 9 11 13
10 12 14 16 18 10 12 14
11 13 11 13
12 12
13 13
14
Figure 6. 𝑔 of A
A=𝟏𝟏,𝟖𝟐,𝟓𝟕,𝟐,𝟏𝟑 → 𝒈𝟔 of A=𝟏𝟏,𝟖𝟐,𝟔𝟑,𝟓,𝟒𝟑,𝟐,𝟏𝟑

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
5
1 5 9 13 17 21 25 29 33 37 41 1 5 9 13 17 21 25 29 33 37 41
2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30
3 7 11 15 19 23 27 31 3 7 11 15 19 23 27 31
4 8 12 16 20 4 8 12 16 20 24
5 9 13 17 21 5 9 13 17 21 25
6 10 14 18 22 6 10 14 18 22 26
7 11 15 19 23 7 11 15 19 23
8 12 16 20 24 8 12 16 20
9 13 17 21 25 9 13 17 21
10 14 18 22 26 10 14 18 22
11 15 11 15
12 12
13 13
14 14
Figure 7. 𝒈𝟔 of A
A=𝟏𝟏,𝟖𝟐,𝟓𝟕,𝟐,𝟏𝟑 → 𝒈𝜶𝟕 of A=𝟏𝟏,𝟖𝟐,𝟔,𝟓𝟓,𝟒,𝟐,𝟏𝟑
1 7 13 19 25 31 37 43 49 55 61 1 7 13 19 25 31 37 43 49 55 61
2 8 14 20 26 32 38 44 2 8 14 20 26 32 38 44
3 9 15 21 27 33 39 45 3 9 15 21 27 33 39 45
4 10 16 22 28 4 10 16 22 28 34
5 11 17 23 29 5 11 17 23 29
6 12 18 24 30 6 12 18 24 30
7 13 19 25 31 7 13 19 25 31
8 14 20 26 32 8 14 20 26 32
9 15 21 27 33 9 15 21 27 33
10 16 22 28 34 10 16 22 28
11 17 11 17
12 12
13 13
14 14
Figure 8. 𝒈𝜶𝟕 of A
It is very clear that by using this new technology, it has solved one of the most basic dilemmas in the
concept of p-regularisation, and according to the new technology we have several possible possibilities
to occur as follows:
Table 3. The New Partition for Each English Letters After Using 𝒈𝜶
𝒈𝟎 𝒈𝟐 𝒈𝟑 𝒈𝟒 𝒈𝟓 𝒈𝟔 𝒈𝜶𝟕
A 𝟏𝟒,𝟏𝟑,𝟏𝟏,𝟗,
𝟖,𝟔,𝟒,𝟐 𝟏𝟏,𝟗,𝟖,𝟕𝟐,
𝟔,𝟓,𝟒,𝟑𝟐,𝟐,𝟏𝟐
- - - 𝟏𝟏,𝟖𝟐,𝟔𝟑,
𝟓,𝟒𝟑,𝟐,𝟏𝟑 𝟏𝟏,𝟖𝟐,𝟔
,𝟓𝟓,𝟒,𝟐,𝟏𝟑
B
- 𝟏𝟑,𝟏𝟐,𝟏𝟏,𝟏𝟎,𝟗,
7,6,5,4,2,1)
- 𝟏𝟐,𝟏𝟏,𝟏𝟎𝟐,𝟖,𝟕
6,𝟓𝟐,𝟑,𝟐,𝟏
- - -
C
- (15,14,12,10,9,
6,5,3,1)
- 𝟏𝟒,𝟏𝟑,𝟏𝟐𝟐,
9,𝟓𝟐,𝟐𝟐,𝟏
- - -
D
- (14,13,11,10,9,
7,6,5,4,2)
- (13,12,𝟏𝟏𝟐,𝟗,𝟖,
6,5,3,2,1)
- - -
E
- - (15,13,11,10,9,
7,6,5,3,1)
- 𝟏𝟑𝟐,𝟏𝟏𝟐,𝟖,𝟕,
6,5,3,2,1)
- -
F
- - (12,9,8,6,
5,3,1)
- (12,8,7,6,
5,3,2,1)
- -
G
- - (14,13,11,10,9,
8,6,5,3,2,1)
- (12,11,𝟏𝟎𝟐,𝟖𝟐,
7,6,5,𝟐𝟐,𝟏
- -
H
- - (14,13,11,10,9,
8,6,5,4,3,1)
- (13,11,10,9,8,
𝟕𝟐,𝟔,𝟓,𝟒,𝟑,𝟏
- -
I
- (17,15,13,
12,8,4,3)
- (16,15,14,12,
8,4,2,1)
- - -
J
- (14,11,10,
8,4,3)
- (14,11,10,
8,4,2,1)
- - -
K
- - (15,13,11,10,8,
6,5,4,3,1)
- NO CHANGE - -
L
- - (20,18,16,
14,13,9,5,1)
- 𝟏𝟖𝟐,𝟏𝟔𝟐,
13,9,5,1)
- -
M
- - (13,12,11,10,
9,7,6,5,3,2)
- NO CHANGE - -
N
- - (15,14,12,11,10,
9,5,4,3,1)
- (11,𝟗𝟐,𝟖,𝟕𝟐,𝟔𝟐, 𝟓𝟐,𝟒,𝟑𝟐,𝟏
- -

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
6
O
- (14,13,11,10,
8,7,6,5,3,1)
- (13,12,𝟏𝟏𝟐,𝟖𝟐, 𝟓𝟐,𝟐𝟐,𝟏
- - -
P
- (13,11,10,9,
8,6,5,4,3,1)
- (12,11,10,8,7,
6,𝟓𝟐,𝟑,𝟐,𝟏
- - -
Q
- - (15,14,12,11,10,
9,8,6,5,3,2)
- 𝟏𝟐𝟐,𝟏𝟏𝟐,𝟗𝟐,
𝟖𝟐,𝟓𝟐,𝟐𝟐,𝟏
- -
R
- (13,11,10,9,8,
7,5,4,3,1)
- (13,11,10,8,7,
6,𝟓𝟐,𝟑,𝟐,𝟏
- - -
S
- (15,14,12,10,
8,6,4,1)
- (14,13,𝟏𝟐𝟐,
𝟕𝟐,𝟐𝟐,𝟏
- - -
T
- (14,10,
7,5,1)
- (14,10,
6,𝟑𝟐,𝟏
- - -
U
- - (15,13,12,10,
9,7,6,4,3,1)
- NO CHANGE - -
V
- - (16,13,12,
11,9,7,5)
- NO CHANGE - -
W
- - (16,14,13,12,10,
9,8,7,6,5)
- NO CHANGE - -
X
- - NO CHANGE - NO CHANGE - -
Y
- (16,12,11,
10,8,6,5)
- (16,12,10,
9,𝟖𝟐,𝟓
- - -
Z
- (17,15,13,12,
10,8,7,5,3)
- (15,14,13,12,11,
10,7,4,𝟐𝟐
- - -
Notes (3.2):
1- When 𝑔 is equal to 2 then any 𝜃 greater than or equal to 2 in original partition will have a
major role in 𝑔,𝜑 regularisation for this case where you will perform a process similar to
decay and natural overlap with each other to achieve the desired property of it and thus will be
the values of new partition all numbers that are not repeated at all, meaning raised to the power
of 1 always.
2- In the case of 𝑔 is equal to 3, then we observe the 𝜃 present in the original partition, each one
of which is greater or equal to 3 is the one that will have the greatest effect in the regularization
process by interfering with the rest and will always be decomposed
into numbers raised to the
power of 2 or the exponent of one. Other than that, it has no effect.
3- Based on the first and second notes, any 𝑔 greater than those cases will have the same path
followed, where 𝜃 equal to the value of 𝑔 or greater will be decomposed into forces that are
smaller than the value of 𝑔 which will have the focus on it and always according to the
conditions of 𝑔,𝜑 that will be adopted for each case studied.
From the above, and through the processes according to which the 𝑔,𝜑-regularisation was
calculated, we notice that we have two regions, the first stable, which is more like a right-angled
isosceles triangle, and the second, turbulent, which tries to be more like the stable through a straight line
parallel to the so-called
diameter of the right-angled triangle, where this straight line will not contain
voids or discontinuities inside it, or to be closer to isolated points, but not far from parallel lines, as
follows:
M 1st step of 𝒈𝟐 of M Final step of 𝒈𝟐 of M
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
2 3 4 5 6 7 8 9 1
0
2 3 4 5 6 7 8 9 1
0
1
1
1
2
2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
3 4 5 6 7 8 9 1
0
1
1
3 4 5 6 7 8 9 1
0
1
1
1
2
3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
4 5 6 7 8 9 1
0
1
1
4 5 6 7 8 9 1
0
1
1
1
2
4 5 6 7 8 9 1
0
1
1
1
2
1
3
5 6 7 8 9 1
0
1
1
5 6 7 8 9 1
0
1
1
1
2
5 6 7 8 9 1
0
1
1
1
2
1
3
6 7 8 9 1
0
1
1
1
2
6 7 8 9 1
0
1
1
1
2
6 7 8 9 1
0
1
1
1
2
7 8 9 1
0
1
1
1
2
7 8 9 1
0
1
1
1
2
7 8 9 1
0
1
1
1
2
8 9 1
0
1
1
1
2
8 9 1
0
1
1
1
2
8 9 1
0
1
1
1
2

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
7
9 1
0
1
1
1
2
1
3
9 1
0
1
1
1
3
9 1
0
1
1
1
0
1
1
1
2
1
3
1
0
1
1
1
3
1
0
1
1
1
1
1
2
1
3
1
3
1
2
1
3
1
3
1
3
1
3
Figure 9. Process stages of 𝒈𝟐 of M
4. Applications
The use of the above technology will be completely beneficial in several other applications, including
the following:
4.1. e-Abacus diagram by James
For any partition of r there exist an diagram; called e-abacus diagram, it was presented by James in [ 7
] to be another type of representation for partition which has made great progress in this area. Fix 𝜔 is
a partition of 𝑟 and defining 𝛽𝜔𝑏𝑖, 1𝑖𝑏. The set 𝛽,𝛽,⋯,𝛽 is said to be the set of 𝛽
- number for 𝜔. Let e be a positive integer number greater than or equal to 2, we can represent numbers
by a diagram called e-abacus diagram, as shown:
Table 4. 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 𝛽 will be represented by a ● and the rest of the sites by (-). In fact, the definition of
e-Abacus diagram will leads us to the fact that the presence of an infinite number of diagrams are all
suitable for any partition according to the value of e. For example, M = 12, 9,8,7
,6,5
, 4,3,2,1 and
e=5, then
- ● - ● -
● - ● - ●
● - ● - ●
● - ● - ●
● - - -
●
Now, if we use the technique of 𝑔 of M; the value of this operation is (13,12,11,10,9,7,6,5,3,2), we
have another e-abacus diagram, as following:
Table 5. e-Abacus Diagram of 𝒈𝟐 of M
- -
● - ●
- - ● - ●
- ● - -
●
- ● - ● -
● - ● - -
It is quite clear that this application is a new type of encoding process for English letters that has not
been used before.
4.2. Replace the content in e-abacus diagram
Mahmood in [8]Introduce a method for replacing content in an e-abacus diagram by replacing each
bead with a slash. So if this method is combined with the regularisation of English letters that was

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
8
presented above, we will have the following stages, which will be applied through at least one example,
let it be:
M
𝟏𝟐,𝟗𝟐,𝟖,𝟕𝟐,𝟔,𝟓𝟐,𝟒,𝟑,𝟐,𝟏
Replace of content
𝟏𝟐𝟑,𝟏𝟎,𝟗,𝟕,𝟔,𝟒,𝟑,𝟐,𝟏
- ● - ● -
● - ● - ●
● - ● - ● -
● - ● -
● - ● - ● -
● - ● -
● - ● - ● -
● - ● -
● - - -
● -
● ● ● -
Now, 𝑔 of the replace of content is:
𝟏𝟐𝟑,𝟏𝟎,𝟗,𝟕,𝟔,𝟒,𝟑,𝟐,𝟏 𝒈𝟐 of 𝟏𝟐𝟑,𝟏𝟎,𝟗,𝟕,𝟔,𝟒,𝟑,𝟐,𝟏
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1 2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
1
4
2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
2 3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
1
4
3 4 5 6 7 8 9 1
0
1
1
1
2
1
3
4 5 6 7 8 9 1
0
1
1
1
2
1
3
4 5 6 7 8 9 1
0
1
1
1
2
1
3
5 6 7 8 9 1
0
1
1
1
2
1
3
5 6 7 8 9 1
0
1
1
1
2
6 7 8 9 1
0
1
1
1
2
6 7 8 9 1
0
1
1
1
2
7 8 9 1
0
1
1
1
2
7 8 9 1
0
1
1
1
2
8 9 1
0
1
1
8 9 1
0
1
1
9 1
0
1
1
9 1
0
1
1
1
0
1
1
1
0
1
1
1
1
1
1
Then we have a new partition is (14,12,11,10,8,7,6,4,3,2,1).
Or 𝑔 then we have another new partition (13,12,11,10,9,7,6,4,3,2,1):
𝟏𝟐𝟑,𝟏𝟎,𝟗,𝟕,𝟔,𝟒,𝟑,𝟐,𝟏 𝒈𝟒 of 𝟏𝟐𝟑,𝟏𝟎,𝟗,𝟕,𝟔,𝟒,𝟑,𝟐,𝟏
1 3 5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
2
2
1 3 5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
2
2
2
4
2 4 6 8 1
0
1
2
1
4
1
6
1
8
2
0
2
2
2
3
2 4 6 8 1
0
1
2
1
4
1
6
1
8
2
0
2
2
2
3
3 5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
2
3
2
4
3 5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
2
3
4 6 8 1
0
1
2
1
4
1
6
1
8
2
0
2
2
4 6 8 1
0
1
2
1
4
1
6
1
8
2
0
2
2
5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
5 7 9 1
1
1
3
1
5
1
7
1
9
2
1
6 8 1
0
1
2
1
4
1
6
1
8
6 8 1
0
1
2
1
4
1
6
1
8
7 9 1
1
1
3
1
5
1
7
7 9 1
1
1
3
1
5
1
7
8 1
0
1
2
1
4
8 1
0
1
2
1
4
9 1
1
1
3
9 1
1
1
3
1
0
1
2
1
0
1
2
1
1
1
1
Therefore we have:

ICMAICT 2020
Journal of Physics: Conference Series 1804 (2021) 012031
IOP Publishing
doi:10.1088/1742-6596/1804/1/012031
9
M
→
Replace the
content
→
𝒈𝟐 of replacement
→
𝒈𝟒 of replacement
- ● - ● - ● - ● - ● - ● - ● - -
● - ● -
● - ● - ● - ● - ● - ● - ● - - ● - ● - -
● - ● - ● - ● - ● - ● - ● - ● ● - ● - -
● - ● - ● - ● - ● - - -
● - ● ● - ● - ●
● - - -
● - ● ● ● - -
● - -
● - ● - ● -
5. Conclusion
1- In this paper, the problem of p-regularization was addressed, which was applicable to small and narrow
cases, with a new method based on taking the largest s in our given partition.
2- We have a set of options regarding regularisation instead of one possible process or not!
3- Using these technologies, we will have new horizons towards encoding of a special type for English
letters that can be used easily on them and not easy to detect at the same time.
4- It is possible to use this type on the e-abacus diagram directly without any complications, and in this case
it can be implemented on all studies that follow the style of this diagram, which will make a wonderful
transfer to this topic, as we only presented two applications, hoping that you will study other cases later.
Acknowledgment: Great thanks and appreciation to the Deanship of the College of Education for Pure
Sciences / University of Mosul for the moral support in carrying out this
paper.
References
[1] Mathas A., Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, University
Lecture Series, vol.15, 1999.
[2] James G. D., On the decomposition matrices of the symmetric groups. II, J. of Algebra, 43(1976),
45-54.
[3] Fayers M., Regularisation and the Mullineux map, Electronic J. of Combinatorics, 15(2008), 1-
15.
[4] Phi Hung T. V., Rank 3 Permutation Characters and Maximal subgroups, Ph. D. Thesis,
University of Birmingham, UK, 2009.
[5] Mahmood A. B. and Mahmood A. S., Secret-word by e-abacus diagram I, Iraqi J. of Science,
60(2019), no. 3, 638-646.
[6] Mahmood A. B. and Mahmood A. S. , Secret-text by e-abacus diagram II, Iraqi J. of Science,
60(2019), no. 4, 840--846.
[7] James G. D., Some combinatorial results involving Young diagrams, Mathematical Proceeding
of the Cambridge Philosophical Society, 83(1978), 1-10.
[8] Mahmood A. S., Replace the content in e-abacus diagram, Open Access Library Journal, 7(2020),
1-6