Nested chain movement of length 1 of beta number in James abacus diagram

• Main Authors: Mohommed, Eman F., Ahmad, Nazihah, Ibrahim, Haslinda, Mahmood, Ammar Seddiq
• Format: Article
• Language: English
• Published: Research India Publications 2016
• Subjects: QA Mathematics
• Online Access: http://repo.uum.edu.my/21545/1/GJPAM%2012%204%202016%202953-2969.pdf; http://repo.uum.edu.my/21545/; http://www.ripublication.com/gjpam16/gjpamv12n4_17.pdf
• Institution: Universiti Utara Malaysia

James abacus diagram is a graphical representation for any partition µ of a positive integer t.One way of producing the diagram is by using beta number with the special number of even columns e, where e � 2.This paper constructs a new method for partition µ with a single motion of nested chain movement of length 1 in James abacus diagram. First, the establishment of an arithmetic sequence among the nested chains of the diagram position is considered.Then, for the movement, we select several beta numbers as the initial points in every chain.The location of the rest of the beta numbers in the James abacus diagram would be changed anticlockwise by length 1 accordingly when the positions of initial beta numbers are changed. Using these new nested chains, a new diagram Atc1 that displays a new partition, is constructed. Furthermore, guides, which are finite number of partitions that are obtained from the original partition after adding zeros, are developed. The number of common beta numbers among these developed guides are then determined. We have established rules to obtain new diagram using a single motion of nested chain movement of length 1.The new diagram can be used in areas of number theory and design.Finally, the proposed method is employed as a special type of James abacus diagram where the number of columns is an even integer smaller than the number of rows in the diagram.