Partitions and elementary symmetric polynomials -- an experimental approach

Given a partition, we write eⱼ () for the j^th elementary symmetric polynomial eⱼ evaluated at the parts of and eⱼpA (n) for the sum of eⱼ () as ranges over the set of partitions of n with parts in A. For eⱼpA (n), we prove analogs of the classical formula for the partition function, p (n) =1/n ₊=₀^n-1₁ (n-k) p (k), where ₁ is the sum of divisors function. We prove several congruences for e₂p₄ (n), the sum of e₂ over the set of partitions of n into four parts. Define the function preⱼ () to be the multiset of monomials in eⱼ (), which is itself a partition. If A is a set of partitions, we define preⱼ (A) to be the set of partitions preⱼ () as ranges over A. If P (n) is the set of all partitions of n, we conjecture that the number of odd partitions in pre₂ (P (n) ) is at least the number of distinct partitions. We prove some results about pre₂ (B (n) ), where B (n) is the set of binary partitions of n. We conclude with conjectures on the log-concavity of functions related to eⱼp (n), the sum of eⱼ () for all P (n).