X1α1…Xnαn where the exponents αi are natural numbers (possibly zero); writing α = (α1,…,αn) this can be abbreviated to X α. The monomial symmetric polynomial mα(X1, …, Xn) is defined as the sum of all monomials xβ where β ranges over all distinct permutations of (α1,…,αn). For instance one has , Clearly mα = mβ when β is a permutation of α, so one usually considers only those mα for which α1 ≥ α2 ≥ … ≥ αn, in other words for which α is a partition of an integer. These monomial symmetric polynomials form a vector space basis: every symmetric polynomial P can be written as a linear combination of the monomial symmetric polynomials. To do this it suffices to separate the different types of monomial occurring in P. In particular if P has integer coefficients, then so will the linear combination. The elementary symmetric polynomials are particular cases of monomial symmetric polynomials: for 0 ≤ k ≤ n one has where α is the partition of k into k parts 1 (followed by n − k zeros). Power-sum symmetric polynomials For each integer k ≥ 1, the monomial symmetric polynomial m(k,0,…,0)(X1, …, Xn) is of special interest. It is the power sum symmetric polynomial, defined as All symmetric polynomials can be obtained from the first n power sum symmetric polynomials by additions and multiplications, possibly involving rational coefficients. More precisely, Any symmetric polynomial in X1, …, Xn can be expressed as a polynomial expression with rational coefficients in the power sum symmetric polynomials p1(X1, …, Xn), …, pn(X1, …, Xn). In particular, the remaining power sum polynomials pk(X1, …, Xn) for k > n can be so expressed in the first n power sum polynomials; for example In contrast to the situation for the elementary and complete homogeneous polynomials, a symmetric polynomial in n variables with integral coefficients need not be a polynomial function with integral coefficients of the power sum symmetric polynomials. For an example, for n = 2, the symmetric polynomial has the expression Using three variables one gets a different expression The corresponding expression was valid for two variables as well (it suffices to set X3 to zero), but since it involves p3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the