On The Intersection of Young's Diagrams Core

Abstract

In this paper, we will introduce the method of intersection of -numbers for any partition  of a non-negative integer r. The results of this intersection are represented and specificated the exactly position according to a "guide value" and a "main diagram". By using the same method we will create a new way for this intersection after finding the core of each "guide".

Replacing the Content in e-Abacus Diagram II

“…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.…”

“…For example, let ( ,1), then the diagrams in tables 2 and 3 hold: ---------------------------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.…”

“…(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 in1 st 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)…”

The Movement of Orbits and Their Effect on the Encoding of Letters in Partition Theory II

“…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 [6].…”