Polynomial identities for partitions

For any partition λ of an integer n , we write λ =< 11, 22, . . . , nn > where mi(λ) is the number of parts equal to i . We denote by r(λ) the number of parts of λ (i.e. r(λ) = ∑n i=1mi(λ) ). Recall that the notation λ ` n means that λ is a partition of n . For 1 ≤ k ≤ N , let ek be the k-th elementary symmetric function in the variables x1, . . . , xN , let hk be the sum of all monomials of total degree k and let pk = ∑ i x k i . Each of the sets {ek} , {hk} or {pk} is a set of algebraically independent generators of the ring of symmetric functions over the field Q . The expressions of hn and en in terms of the pk are the following [Mac, page 17]: