Then the existence of a doubly stochastic matrix P such that a = Pb is equivalent to the following system of inequalities: (The last one is an equality; the others are weak inequalities.) The sequence is said to majorize the sequence . Symmetric sum notation It is convenient to use a special notation for the sums. A success in reducing an inequality in this form means that the only condition for testing it is to verify whether one exponent sequence () majorizes the other one. This notation requires developing every permutation, developing an expression made of n! monomials, for instance: Examples Arithmetic-geometric mean inequality Let and We have Then [aA] ≥ [aG], which is yielding the inequality. Other examples We seek to prove that x2 + y2 ≥ 2xy by using bunching (Muirhead's inequality). We transform it in the symmetric-sum notation: The sequence (2, 0) majorizes the sequence (1, 1), thus the inequality holds by bunching. Similarly, we can prove the inequality by writing it using the symmetric-sum notation as which is the same as Since the sequence (3, 0, 0) majorizes the sequence (1, 1, 1), the inequality holds by bunching. See also Inequality of arithmetic and geometric means Doubly stochastic matrix Monomial symmetric polynomial Notes References Combinatorial Theory by John N. Guidi, based on lectures given by Gian-Carlo Rota in 1998, MIT Copy Technology Center, 2002. Kiran Kedlaya, A < B (A less than B), a guide to solving inequalities Hardy, G.H.; Littlewood, J.E.; Pólya, G. (1952), Inequalities, Cambridge Mathematical Library (2. ed.), Cambridge: Cambridge University Press, , , , Section 2.18, Theorem 45. Inequalities Means