Let r be a non-negative integer. A partition of r is a sequence of non-negative integers such that and for , . Fix is a partition of r, choosing an integer b greater than or equal to the number of parts of and defining . The set is said to be the set of -number for , see   .
Let e be a positive integer number greater than or equal to 2, we can represent -number by a diagram called e-Abacus diagram: (Table 1).
Where every will be represented by a bead (●) and else that by (-) which takes its location in Table 1.
A formula was adopted in  for the format of the orbits for any English letter,
Table 1. e-Abacus diagram.
Figure 1. When the word of 2 letters.
Figure 2. When the word of 3 letters.
we have three orbit according to 3.1; only the case of 2-orbit is discussed there because it is the one that has the most influence in that research and the rest. Now, if we have a word of 2, 3 or more letters. Will we try encoding on each letter separately or we will use the encoding on each word? And because the process of calculating the partition is based on all the word, see   , we had to find a mechanism to encode the word according to the movement of the orbits.
2.1. Behavior of Each Orbit to Any Word
Obviously there are three orbits:
1) Orbit: It is the external orbit which will remain the same without any change so that we can read the partition of the word before the change, see  .
2) Orbit: It is the middle orbit which takes the following location.
3) Orbit: It is the last orbit and his movement will be explained in section 3 of this paper.
2.2. for Any Word
Thus we can make the following rules:
Proof: Since the transition process will be in two ways and each of them has two opposite directions. The first way (with traditional direction) will be taken when , as for the opposite direction of the same way, it is when . Now, we come to the second way (the traditional direction), it will be when , and for last direction then we have when .
For example (Figure 3),
Will be (Figure 4):
Proof: By using the same method of rule (2.2.1) we have 4 ways (with two opposite directions) as the following:
If (5ℏ − 1) is odd, then:
If (5ℏ − 1) is even, then:
Then all the relationships above are achieved.
For example the word (Figure 5).
(I) If w2 = 3 then
(II) If w2 = 4 then
Figure 5. The word WAY when .
(III) If w2 = 5 then:
(IV) If w2 = 6 then we have:
(V) If w2 = 7 then we have:
For example the word (Figure 6).
Figure 6. The word WAY when .
Figure 7. The positions in orbit 3 for any word have 2 letters.
Figure 8. The positions in orbit 3 for any word have 3 letters.
Figure 9. The word WAY when .
Figure 10. The word WAY when .
3. The Movement of w3
By the results of  , mentioned that per letter has no effect as only one position, but in the case of a word consisting of more than one letter, its impact is very important. On this basis in the case of a word that contains only two letters, then (Figure 7, Figure 8).
Rule 3.1: When choosing a partition for any word have ( ) letter where e = 5 and the value of was equal to the location within will be:
4. Results and Discussion
1) This encoding made the first encoding of English letters more difficult in terms of finding the origin of the word.
2) A regular shape was used from e-Abacus diagram and we can think of using an irregular shape in the future.
3) It is quite possible to merge both and at the same time by merging the previous relationships with each other.
1) The above technique can be used on letters of other languages that do not use the same letters.
2) This technique can be used in tiling, were the colors and shapes vary.
We extend our thanks and appreciation to the University of Mosul for their great support for the completion of the research.
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