Abstract:

Let p(n) denote the number of unrestricted partitions of n. For
i=0, 2, let p[i](n) denote the number of partitions pi of n
such that O(pi) - O(pi') = i mod 4.
Here O(pi) denotes the number of odd parts of the partition pi
and pi' is the conjugate of pi. 
Recently, R. Stanley [13], [14] derived an infinite product expansion
for the generating function of p[0](n)-p[2](n). 
Recently, H. Swisher[15] employed the circle method to show that

(i)  limit[n->oo] p[0](n)/p(n) = 1/2

and that for sufficiently large n

     2 p[0](n) > p(n), if n=0,1 mod 4,
(ii)
     2 p[0](n) < p(n), otherwise.

In this paper we study the even/odd dissection of the Stanley product, and
show how to use it to prove (i) and (ii) with no restriction on n. Moreover, 
we establish the following new result

|p[0](2n) - p[2](2n)| > |p[0](2n+1) - p[2](2n+1)|, n>0.

Two proofs of this surprising inequality are given.  The first one uses the
Gollnitz-Gordon partition theorem. The second one is an immediate corollary
of a new partition inequality, which we prove in a combinatorial manner. Our
methods are elementary. We use only Jacobi's triple product identity and some
naive upper bound estimates.

The url of this page is http://www.math.ufl.edu/~frank/abstracts/stanley3.html.
Created by Francis Garvan (frank@math.ufl.edu) on Saturday, September 25, 2004.
Last update made Mon Dec 19 19:33:28 EST 2005.

frank@math.ufl.edu