Research On Combinatorial Properties Of Symmetric Functions And Their Applications

Symmetric function theory is one of the most important research fields in alge-braic combinatorics, it mainly studies the algebraical and combinatorial propertiesof symmetric groups and symmetric polynomial. Here we study the combinato-rial properties and applications of three particular symmetric functions: elementarysymmetric function, complete symmetric function and power sum symmetric func-tion. The content is as follows:Chapter 1 of this dissertation is devoted to introducing the three primary re-search objects of this paper: elementary symmetric function, complete symmetricfunction and power sum symmetric function. Their definition, basic properties andthe research situation are introduced. Finally their combinatorial interpretationsare given.In Chapter 2, several new properties of three particular symmetric functions andrelationships among them are obtained, meanwhile some identities which involvingbinomial coe?cients and Stirling numbers are obtained.In Chapter 3, the factorizations of the elementary symmetric polynomial matrixand complete symmetric polynomial matrix are obtained by using the recurrencerelations among their elements. The factorizations of Vandermonde matrix and itsinverse are obtained by the elementary symmetric polynomial matrix and completesymmetric polynomial matrix. Furthermore a connection between the inverse andtranspose of the Vandermonde matrix is established.