Partitions and elementary symmetric polynomials -- an experimental approach

Given a partition λ, we write eⱼ(λ) for the jᵗʰ elementary symmetric polynomial eⱼ evaluated at the parts of λ and eⱼpA(n) for the sum of eⱼ(λ) as λ 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 ∑ₖ₌₀ⁿ⁻¹σ₁(n-k)p(k), where σ₁ 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(λ) to be the multiset of monomials in eⱼ(λ), which is itself a partition. If A is a set of partitions, we define preⱼ( A) to be the set of partitions preⱼ(λ) as λ 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(λ) for all λ∈ P(n).