elementary symmetric polynomials that have degree (they are in fact homogeneous) as follows by partitions of . Order the individual elementary symmetric polynomials in the product so that those with larger indices come first, then build for each such factor a column of boxes, and arrange those columns from left to right to form a Young diagram containing boxes in all. The shape of this diagram is a partition of , and each partition of arises for exactly one product of elementary symmetric polynomials, which we shall denote by ) (the is present only because traditionally this product is associated to the transpose partition of ). The essential ingredient of the proof is the following simple property, which uses multi-index notation for monomials in the variables . Lemma. The leading term of is . Proof. The leading term of the product is the product of the leading terms of each factor (this is true whenever one uses a monomial order, like the lexicographic order used here), and the leading term of the factor is clearly . To count the occurrences of the individual variables in the resulting monomial, fill the column of the Young diagram corresponding to the factor concerned with the numbers of the variables, then all boxes in the first row contain 1, those in the second row 2, and so forth, which means the leading term is . Now one proves by induction on the leading monomial in lexicographic order, that any nonzero homogeneous symmetric polynomial of degree can be written as polynomial in the elementary symmetric polynomials. Since is symmetric, its leading monomial has weakly decreasing exponents, so it is some with a partition of . Let the coefficient of this term be , then is either zero or a symmetric polynomial with a strictly smaller leading monomial. Writing this difference inductively as a polynomial in the elementary symmetric polynomials, and adding back to it, one obtains the sought for polynomial expression for . The fact that this expression is unique, or equivalently that all the products (monomials) of elementary symmetric polynomials are linearly independent, is also easily proved. The lemma shows that all these products have different leading monomials, and this suffices: if a nontrivial linear combination of the were zero, one focuses on the contribution in the linear combination with nonzero coefficient and with (as polynomial in the variables ) the largest leading monomial;