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

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.