Cyclic Tableaux and Symmetric Functions

Abstract

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function pn in terms of the elementary symmetric function en and the homogeneous symmetric function hn. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.