Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/22

${\begin{aligned}h_{1}&=s_{1}{\text{,}}\\2!\,h_{2}&={s_{1}}^{2}+s_{2}{\text{,}}\\3!\,h_{3}&={s_{1}}^{3}+3s_{1}s_{2}+2s_{3}{\text{,}}\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\n!\,h_{n}&={\textstyle \sum {}}{\frac {n!}{1^{\pi _{1}}.2^{\pi _{2}}\ldots k^{\pi _{k}}.\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{s_{1}}^{\pi _{1}}{s_{2}}^{\pi _{2}}\ldots {s_{k}}^{\pi _{k}}{\text{.}}\end{aligned}}$

These are the principal properties of symmetric functions that will be of use.

9. If we take any assemblage of letters such as $\alpha \alpha \beta \gamma$ and are not concerned with the order in which these letters are written, we have a ‘Combination’ of the letters. If however the order in which the letters are written be taken into account, we have a ‘Permutation’ of the letters. In the present case we have twelve permutations, viz. ${\begin{aligned}&\alpha \alpha \beta \gamma \quad \alpha \alpha \gamma \beta \quad \alpha \beta \alpha \gamma \quad \alpha \beta \gamma \alpha \quad \alpha \gamma \alpha \beta \quad \alpha \gamma \beta \alpha \\&\beta \alpha \alpha \gamma \quad \beta \alpha \gamma \alpha \quad \beta \gamma \alpha \alpha \quad \gamma \alpha \alpha \beta \quad \gamma \alpha \beta \alpha \quad \gamma \beta \alpha \alpha \\\end{aligned}}$

10. In a similar manner if we take any collection of integers which add up to a given integer we have as above defined (Art. 3) a partition of the given number; here no account is taken of the order in which the parts of the partition may be written; but if order has to be taken into account each way of writing the parts is called a ‘Composition’ of the number, such composition appertaining to the particular partition which is involved. Thus of the number $9$, $3321\equiv 3^{2}21$ is a partition which gives rise to the twelve compositions: ${\begin{aligned}&3321\quad 3312\quad 3231\quad 3213\quad 3132\quad 3123\\&2331\quad 2313\quad 2133\quad 1332\quad 1323\quad 1233\\\end{aligned}}$ and it will be noticed that the compositions which appertain to the partition $3321$ of the number $9$ are in correspondence with the permutations of the combination $\alpha \alpha \beta \gamma$.

Moreover, in general the compositions which appertain to any given partition of a number are in correspondence with the permutations of a certain combination of letters.

11. In pursuing the main object of this book, namely the study of the algebra of symmetric functions together with those theories of combination, permutation, arrangement, order and distribution which are summed up in the title ‘Combinatory Analysis,’ it is important to have some specific rules for arranging the order in which the terms of algebraic expressions are written down.