20.
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 . Here 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
p0(
n)-
p2(
n). Recently, Swisher [The Andrews–Stanley partition function and
p(
n), preprint, submitted for publication] employed the circle method to show that
and that for sufficiently large
n 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:
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.
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