Dissecting the Stanley partition function

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 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 . 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 (i) lim n → ∞ p 0 ( n ) p ( n ) = 1 2 and that for sufficiently large n (ii) 2 p 0 ( n ) > p ( n ) if n ≡ 0 , 1 ( mod 4 ) , 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 ( 2 n ) - p 2 ( 2 n ) | > | p 0 ( 2 n + 1 ) - p 2 ( 2 n + 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.