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.

Keywords

• Compositions of positive integers
• partitions of positive integers
• the odd combination set of an integer
• Fibonacci numbers
• generating function

References

1. Al, B., Alkan, M., Some relations between partitions and Fibonacci numbers, Proc. Book of 2nd. Micopam, (2019).
2. Al, B., Alkan, M., On relations for the partitions of numbers, Filomat, 34(2)(2020), 567–574.
3. Al, B., Alkan, M., Note on non-commutative partition, Proc. Book of 3nd&4th. Micopam, (2020-2021).
4. Al, B., Alkan, M., Odd compositions and odd partititons on positive integers, Proc. Book of 5th. Micopam, (2022).
5. Al, B., Alkan, M., Bir tam sayının parça sayısıyla kısıtlanmış kompozisyonları, Proc. Book of ICEANS, (2022).
6. Al, B.,Alkan, M., Compositions of an integers and Fibonacci numbers, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, (2023) (accepted).
7. Andrews, G.E., The Theory of Partitions, Addison-Wesley Publishing, New York, 1976.
8. Andrews, G.E., Erikson, K., Integer Partitions, Cambridge University Press, Cambridge, 2004.

9. Andrews, G.E., Hirschhorn, M.D., Sellers, J. A., Arithmetic properties of partitions with even parts distinct, Ramanujan Journal,23(1–3)(2010), 169–181.
10. Apostol, T.M., On the Lerch Zeta function, Pacific J. Math., 1(1951), 161–167.
11. Apostol, T.M., Introduction To Analytic Number Theory, Springer-Verlag, NewYork, 1976.
12. Chen, S.C., On the number of partitions with distinct even parts, Discrete Math., 311(2011), 940–943.
13. Euler, L., Introduction to Analysis of the Infinite, vol. 1, Springer-Verlag, 1988 (translation by J.D. Blanton).
14. Ewell, J.A., Recurrences for the partition function and its relatives, Rocky Mountain Journal Of Mathematics, 34(2)(2004).
15. Ewell, J.A., Recurrences for two restricted partition functions, Fibonacci Quart., 18(1980), 1–2.
16. Gupta, H., Partitions - A Survey, Journal of Research of the Notional Bureau of Standards - B. Mathematical Sciences, 74B(1)(1970).
17. Hardy, G.H., Wright, E.M., An Introduction to the Theory of Numbers, 4th ed., Clarendon Press, Oxford, 1960.
18. Horadam, A.F., Jacobsthal representation numbers, Fibonacci Quarterly, 34(1)(1996), 40–54.
19. Koshy, T., Fibonacci and Lucas Numbers with Applications, Canada: Wiley-Interscience Publication, 2001.
20. MacMahon, P.A., Note on the parity of the number which enumerates the partitions of a number, Proc. Cambridge Philos. Soc., 20(1921), 281–283.
21. Mana, P., Problem B-152, The Fibonacci Quarterly, 7(3)(1969), 336.
22. Merca, M., Fast computation of the partition function, Journal of Number Theory, 164(2016,) 405–416.
23. Merca, M., New relations for the number of partitions with distinct even parts, Journal of Number Theory, 176(2017), 1–12.
24. Merca, M., On the number of partitions into parts of k different magnitudes, Discrete Mathematics, 340(2017,) 644–648.
25. Merca, M., A note on the partitions involving parts of k different magnitudes, Journal of Number Theory, 162(2016), 23–34.