Symmetric Functions of a Two-way Array

Abstract

A family of polynomials in the elements of a two-way array, or matrix, is introduced. This family is an extension, from sets to matrices, of the family of symmetric polynomials $k_1, k_2, k_{11}, k_3, k_{12}$, etc., defined by Tukey [6], christened "polykays" in [7], and which are a generalization of the family $k_1, k_2$, etc., defined by R. A. Fisher [1]. The polynomials of the present paper, called "bipolykays," are symmetric functions in the sense that they are invariant under permutation of rows and/or columns of the matrix. This paper defines the bipolykays, shows that they are inherited on the average, develops the formulas for use in random pairing, and provides tables for conversion and for multiplication. A description of applications (see [2], [3], and [4]) will be postponed until a later paper. These applications include (a) finding expressions for sampling moments of functions of the elements of a matrix which is a "bi-sample" from a larger matrix, (b) finding expressions for sampling moments of functions (such as estimates of variance components) associated with the analysis of variance of a two-way table with systematic interactions, and (c) finding unbiased estimators for the variances and covariances of estimated variance components in a two-way table without interactions.