Given a partition \(\lambda \) , we write \(e_j(\lambda )\) for the \(j^{\text {th}}\) elementary symmetric polynomial \(e_j\) evaluated at the parts of \(\lambda \) and \(e_jp_A(n)\) for the sum of \(e_j(\lambda )\) as \(\lambda \) ranges over the set of partitions of n with parts in A. For \(e_jp_A(n)\) , we prove analogs of the classical formula for the partition function, \(p(n)=1/n \sum _{k=0}^{n-1}\sigma _1(n-k)p(k)\) , where \(\sigma _1\) is the sum of divisors function. We prove several congruences for \(e_2p_4(n)\) , the sum of \(e_2\) over the set of partitions of n into four parts. Define the function \(\text {pre}_j(\lambda )\) to be the multiset of monomials in \(e_j(\lambda )\) , which is itself a partition. If \({\mathcal {A}}\) is a set of partitions, we define \(\text {pre}_j({\mathcal {A}})\) to be the set of partitions \(\text {pre}_j(\lambda )\) as \(\lambda \) ranges over \({\mathcal {A}}\) . If \({\mathcal {P}}(n)\) is the set of all partitions of n, we conjecture that the number of odd partitions in \(\text {pre}_2({\mathcal {P}}(n))\) is at least the number of distinct partitions. We prove some results about \(\text {pre}_2({\mathcal {B}}(n))\) , where \({\mathcal {B}}(n)\) is the set of binary partitions of n. We conclude with conjectures on the log-concavity of functions related to \(e_jp(n)\) , the sum of \(e_j(\lambda )\) for all \(\lambda \in {\mathcal {P}}(n)\) .
