Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i = 0, 2, let pi(n) denote the number of partitions π of n such that O(π)−O(π) ≡ i (mod 4). Here O(π) denotes the number of odd parts of the partition π and π is the conjugate of π. R. Stanley [13], [14] derived an infinite product representation for the generating function of p0(n)− p2(n). Recently, H. Swisher [15] employed the circle method to show that (i) lim n→∞ p0(n) p(n) = 1 2 , and that for sufficiently large n (ii) 2p0(n) > p(n), if n ≡ 0, 1 (mod 4), 2p0(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 |p0(2n) − p2(2n)| > |p0(2n+ 1)− p2(2n+ 1)|, n > 0. Two proofs of this surprising inequality are given. The first one uses the Göllnitz-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.