Abstract
In 1995, R. Fokkink, W. Fokkink and Wang defined the in terms of , where is the smallest part of partition . In 2008, Andrews obtained the generating function for . In 2013, Andrews, Garvan and Liang extended the FFW-function and obtained the similar expressions for the spt-function and then defined the spt-crank generating functions. They also defined the generating function for in various ways. This paper shows how to find the number of partitions of n into distinct parts with certain conditions and shows how to prove the Theorem 1 by induction method. This paper shows how to prove the Theorem 2 with the help of two generating functions.
Highlights
In this paper we give some related definitions of P(n), FFW n, d(n), (x) (x2; x ),(x)k and(xk 1; x)
We give two tables for FFW(5) and FFW(6) respectively and discuss the generating functions for FFW(n) and shows a relation related to the term d n .We discuss the various generating functions for FFW(z, n) and prove the Corollary I for proving the fundamental Theorem 1 containing three parts and prove the Theorem 2
In this study we have found the number of partitions of n into distinct parts with required conditions
Summary
In this paper we give some related definitions of P(n), FFW n , d(n), (x) (x2; x ) , (zx) ,(x)k and(xk 1; x). They defined the generating function for FFW z, n in various ways. This paper shows how to find the number of partitions of n into distinct parts with certain conditions and shows how to prove the Theorem 1 by induction method. This paper shows how to prove the Theorem 2 with the help of two generating functions.
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