Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/16

The function $\textstyle \sum {\alpha ^{i}}$ is the sum of the $i$th powers of the quantities. It takes a leading part in the algebra of the functions.

The laws of this algebra do not depend upon the absolute magnitudes of the quantities $\alpha ,\beta ,\gamma ,\ldots \nu$, so that usually it is not necessary to specify these quantities. Various notations have been adopted with the object of eliminating the actual magnitudes from consideration. Thus $\textstyle \sum {\alpha ^{i}}$ is sometimes denoted by $s_{i}$; meaning thereby the sum of the $i$th powers of magnitudes which it is not needful to specify either in magnitude or (very often) in number. Others realising that in the algebra they have to deal entirely with the number $i$ have denoted the same function by $(i){\text{,}}$ viz. the number $i$ in round brackets. This notation is of the greater importance because, as will become evident, it can be extended readily to rational and integral functions in general. Not only so; it is fundamentally important because it supplies the connecting link between the algebra of symmetric functions and theories which deal with numbers only and not with algebraic quantities.

2. Proceeding to functions whose representative terms involve two quantities, the simplest we find to be $\textstyle \alpha \beta +\alpha \gamma +\beta \gamma +\ldots +\mu \nu {\text{,}}$ which involves each of the ${\scriptstyle {\frac {1}{2}}}n(n-1)$ combinations, two together, of the $n$ quantities. It is visibly symmetrical.

This is denoted in conformity with the conventional notation by

	$\textstyle \sum {\alpha \beta }{\text{,}}$
or by	$(11){,}$

the function being completely given when $n$ is known.

Every function is considered to have a weight, which is equal to the sum of the numbers that, in the last notation, appear in the brackets.

Thus the functions $(i)$, $(11)$ have the weights $i$, $2$ respectively. When a number is repeated in brackets it is convenient to use repetitional exponents. Thus $(11)$ is frequently written in the form $(1^{2})$.

Of the weight one we have the single function $(1){\text{;}}$ of the weight two, the two functions $(2){\text{,}}\ (1^{2}){\text{.}}$