A Note on the Composition of a Positive Integer whose Parts are Odd Integers

Authors : Busra Al, Mustafa Alkan

Pages : 423-432

Doi:10.47000/tjmcs.1166566

View : 72 | Download : 50

Publication Date : 2023-12-31

Article Type : Research Paper

Abstract :In this study, we interested in the compostions of integers. Then, the combinations of an integer whose each part is odd were examined. \\begin{equation*} O_{n}=\\{(2a_{1}+1,...,2a_{t}+1):\\text{ }2a_{1}+1+...+2a_{t}+1=n\\text{ and \\ } a_{i}\\text{ positive integer}\\}. \\end{equation*} and we call the set as an odd combination set $O_{n}$ set of an integer $n$. Then, an action on the set are defined. Then, the decomposition of the composition sets of a positive integer has been examined by using set theory. Then, we also focused on the combination of an integer n whose sum is less than a fixed integer m. We have obtained the composition set of an integer whose largest part is less than m. Using these sets, we obtained recurrence relations.
Keywords : Compositions of positive integers, partitions of positive integers, the odd combination set of an integer, Fibonacci numbers, generating function

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