Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/15

Chapter I

Elementary theory of Symmetric Functions

1. A great part of Combinatory Analysis may be based upon the algebra of Symmetric Functions, and it is therefore necessary to have some clear definitions and simple properties of such functions before us.

An algebraic function of a number of numerical magnitudes is said to be Symmetrical if it be unaltered when any two of the magnitudes are interchanged. In algebra such magnitudes (or quantities) are denoted by letters of the alphabet.

Restricting ourselves to those functions which are rational it is clear, for example, that the simple sum of the quantities $\alpha ,\beta ,\gamma ,\ldots \nu$, $n$ in number, is such a function. For the sum $\textstyle \alpha +\beta +\gamma +\ldots +\nu$ is unaltered when any selected pair of the letters is interchanged. For this symmetric function, of which $\alpha$ is the type, we adopt the shorthand $\textstyle \sum {\alpha }$.

Again, another symmetric function is $\textstyle \alpha ^{i}+\beta ^{i}+\gamma ^{i}+\ldots +\nu ^{i}$ because the enunciated conditions of symmetry are just as clearly satisfied as in the particular case $i=1$.

We may denote this function by $\textstyle \sum {\alpha ^{i}}$, the representative or typical term being alone put in evidence. This last expression includes all the integral symmetric functions, the representative term of which involves one only of the quantities. If we are not restricted to integral functions the representative term may be any rational function of a. For example $\textstyle \sum {\displaystyle {\frac {\alpha ^{s}}{1-a\alpha ^{i}}}}={\displaystyle {\frac {\alpha ^{s}}{1-a\alpha ^{i}}}+{\frac {\beta ^{s}}{1-a\beta ^{i}}}+{\frac {\gamma ^{s}}{1-a\gamma ^{i}}}+\ldots +{\frac {\nu ^{s}}{1-a\nu ^{i}}}}$, but we are, in most cases, concerned with the symmetric functions which are integral as well as rational.

M. 1