Abstract
In his paper, “On a partition function of Richard Stanley,” George Andrews proves a certain partition identity analytically and asks for a combinatorial proof.This paper provides the requested combinatorial proof.
Highlights
In [6], Stanley posed a problem on partitions
In [2], Andrews studies the function S(r, s; n) which equals the number of partitions π of n such that π has r odd parts and the conjugate π of π has s odd parts, and proves a result from which he solves Stanley’s problem as a corollary
Is the generating function for partitions into odd parts, where each part is counted with weight y and each distinct integer of odd multiplicity is counted with weight x
Summary
In [6], Stanley posed a problem on partitions. In [2], Andrews studies the function S(r , s; n) which equals the number of partitions π of n such that π has r odd parts and the conjugate π of π has s odd parts, and proves a result from which he solves Stanley’s problem as a corollary.Andrews states some additional interesting corollaries, including the following theorem. ∞ n,r ,sS(r , s; n)zr ys qn = j=1 1 − q4j1 + y zq2j−1 1 − z2q4j−2 1 − y 2q4j−2 . (1.1)He proves this theorem analytically as the limiting case of a certain polynomial identity [2, Theorem 1]. In [2], Andrews studies the function S(r , s; n) which equals the number of partitions π of n such that π has r odd parts and the conjugate π of π has s odd parts, and proves a result from which he solves Stanley’s problem as a corollary. I will follow Stanley [6] and Andrews [2] and let ᏻ(π ) denote the number of odd parts in the partition π .
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