CYCLIC TABLEAUX AND SYMMETRICAL FUNCTIONS

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p(n) in terms of the elementary symmetric function e(n) and the homogeneous s ymmetric function h(n). The coefficients appearing in Waring's formula s are shown to be a cyclic analog of the multinomial coefficients, a f act that seems to have been neglected before. Our involutions also spe ll out the duality between these two forms of Waring's formulas, which turns out to be exactly the ''duality between sets and multisets.'' W e also present an involution for permutations in cycle notation, leadi ng to probably the simplest combinatorial interpretation of the Mobius function of the partition lattice and a purely combinatorial treatmen t of the fundamental theorem on symmetric functions, This paper;is mot ivated by Chebyshev polynomials in connection with Waring's formula in two variables.