# Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/18

4

Elementary theory of Symmetric Functions

4. A second important set is constituted by those functions which are denoted by partitions in which only unity appears as a part. It is

 ${\displaystyle (1),(1^{2}),(1^{3}),\ldots (1^{n}),}$ or ${\displaystyle \textstyle \sum {\alpha _{1}},\ \sum {\alpha _{1}\alpha _{2}},\ \sum {\alpha _{1}\alpha _{2}\alpha _{3}},\ \ldots \sum {\alpha _{1}\alpha _{2}\alpha _{3}\ldots \alpha _{n}}{\text{.}}}$

These are sometimes called unitary functions. The set is particularly connected with the Theory of Algebraic Equations because

${\displaystyle \textstyle (x-\alpha )(x-\beta )(x-\gamma )\ldots (x-\nu )}$

${\displaystyle \textstyle =x^{n}-\sum {\alpha .x^{n-1}}+\sum {\alpha \beta .x^{n-2}}-\sum {\alpha \beta \gamma .x^{n-3}}+\ldots }$

the last term being ${\displaystyle \textstyle \pm \sum {\alpha \beta \gamma \ldots \nu }}$, according as ${\displaystyle n}$ is even or uneven. Hence considering the equation

${\displaystyle x^{n}-a_{1}x^{n-1}+a_{2}x^{n-2}-a_{3}x^{n-3}+\ldots +(-)^{n}a_{n}=0}$
and supposing the ${\displaystyle n}$ roots to be
${\displaystyle \alpha ,\beta ,\gamma ,\ldots \nu {\text{,}}}$
it is clear that
${\displaystyle \textstyle x^{n}-a_{1}x^{n-1}+a_{2}x^{n-2}-a_{3}x^{n-3}+\ldots +(-)^{n}a_{n}}$

${\displaystyle \textstyle =x^{n}-\sum {\alpha .x^{n-1}}+\sum {\alpha \beta .x^{n-2}}-\ldots +(-)^{n}\alpha \beta \gamma \ldots \nu {\text{,}}}$

and we at once deduce the relations

{\displaystyle \textstyle {\begin{aligned}a_{1}=&\sum {\alpha }{\text{,}}\\a_{2}=&\sum {\alpha \beta }{\text{,}}\\a_{3}=&\sum {\alpha \beta \gamma }{\text{,}}\\\cdots &\cdots \cdots \cdots \\a_{n}=&\sum {\alpha \beta \gamma \ldots \nu }{\text{.}}\end{aligned}}}
These functions are frequently called ‘elementary’ symmetric functions because they arise in this simple manner.

It is sometimes convenient, undoubtedly, to regard the quantities ${\displaystyle \alpha ,\beta ,\gamma \ldots \nu }$ as being the roots of an equation, the left-hand side of which involves the elementary functions with alternately positive and negative signs, but the notion is not essential to the study of the subject of symmetric functions.

5. There is a third important series of functions.

Of the weight ${\displaystyle w}$ there are functions which in the partition notation are denoted by partitions of the number ${\displaystyle w}$.

There is one function corresponding to every such partition.

Such a function, since it is denoted by a single partition, is called a Monomial Symmetric Function.