Partitions and elementary symmetric polynomials: an experimental approach

Abstract Given a partition $$\lambda $$ λ , we write $$e_j(\lambda )$$ e j ( λ ) for the $$j^{\text {th}}$$ j th elementary symmetric polynomial $$e_j$$ e j evaluated at the parts of $$\lambda $$ λ and $$e_jp_A(n)$$ e j p A ( n ) for the sum of $$e_j(\lambda )$$ e j ( λ ) as $$\lambda $$ λ ranges over the set of partitions of n with parts in A. For $$e_jp_A(n)$$ e j p A ( n ) , we prove analogs of the classical formula for the partition function, $$p(n)=1/n \sum _{k=0}^{n-1}\sigma _1(n-k)p(k)$$ p ( n ) = 1 / n ∑ k = 0 n - 1 σ 1 ( n - k ) p ( k ) , where $$\sigma _1$$ σ 1 is the sum of divisors function. We prove several congruences for $$e_2p_4(n)$$ e 2 p 4 ( n ) , the sum of $$e_2$$ e 2 over the set of partitions of n into four parts. Define the function $$\text {pre}_j(\lambda )$$ pre j ( λ ) to be the multiset of monomials in $$e_j(\lambda )$$ e

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