Dissecting the Stanley partition function

摘要

Let p(n) denote the number of unrestricted partitions of n. For i = 0, 2, let pi (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. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p(0)(n) - p(2)(n). Recently, Swisher [The Andrews-Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that lim(n ->infinity) p(0)(n)/p(n) = 1/2 and that for sufficiently large n 2p(0)(n) > p(n) if n = 0, 1 (mod 4), 2p(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: vertical bar p(0)(2n) - p(2)(2n)vertical bar > vertical bar p(0)(2n + 1) - p(2)(2n + 1)vertical bar, 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.