Some Restricted Partition Functions

Sabuj Das^1,

¹Senior Lecturer, Department of Mathematics Raozan University College, Bangladesh

Cite this paper

Sabuj Das. Some Restricted Partition Functions. American Journal of Applied Mathematics and Statistics. 2014; 2(6):386-393. doi: 10.12691/AJAMS-2-6-5

Abstract

In 1742, Euler found the generating function for P(n). Hardy said Ramanujan was the first, and upto now the only, Mathematician to discover any such properties of P(n). In 1952, Macmahon established a table of P(n) for the first 200 values of n. This paper showed how to find the number of partition of n by using Macmahon’s table. In 1742, Euler also stated the series in the enumeration of partitions. This Paper showed how to generate the Euler’s use of series in the enumeration of partitions. In 1952, Macmahon also quoted the self-conjuate partitions of n. In this Paper, Macmahon’s self-conjugate partitions are explained with the help of array of dots. This paper showed how to prove the Euler’s Theorems with the help of Euler’s device of the introduction of a second parameter z, and showed how to prove the Theorem 3 with the help of Euler’s generating function for P(n), and also showed how to prove the Theorem 4 with the help of certain conditions of P(n).

Keywords

at most, array, convenient, discover, denominator, diagonal, Euler’s device, Euler’s identity, Macmahon’s table, Self-conjugate

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Andrews, GE, An Introduction to Ramanujan’s Lost Notebook, Amer. Math. Monthly, 86, 1979. 89-108.
[2]	Burn, R.P. (1964). A Pathway into Number Theory, 2nd Edition.
[3]	Hardy G.H and Wright, E.M. Introduction to the Theory of Numbers, 4th Edition, Oxford, Clarendon Press, 1965.
[4]	IVAN and NIVEN, An introduction to the theory of Numbers, Fifth edition (1969).
[5]	Macmahon, Combinatory analysis, Ann Arbor, Michigan, University of Michigan Library, 2005.
[6]	S.BARNARD and J.M. CHILD, Higher Algebra, Macmillan (1967).
[7]	S.Ramanujan (1919), Some properties of P(n),number of partitions of n, Proc. of the Cam. Philo. Society X1X: 207-210.
[8]	Sabuj Das and Haradhan Kumer Mohajan, Generating Functions For X(n) an Y(n), American Review of Mathematics and Statistics, Vol. 2 No. 1, March 2014.