Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/17

Observe that two functions present themselves because two objects can either be taken in one lot comprising both objects, or in two lots, one object in each lot. We express this by saying that the number $2$ has two partitions. We have thus, of the weight two, a function corresponding to each partition of $2$.

3. In the notation of the Theory of the Partition of Numbers the partitions of the number $2$ are denoted by $(2)$, $(1^{2})$. It is for this reason that the notation we are employing for symmetric functions is termed ‘The Partition Notation’. Similarly in correspondence with the three partitions of $3$, viz. $(3)$, $(21)$, $(1^{3})$, we have the symmetric functions

$$\textstyle \sum {\alpha ^{3}}{\text{,}}\sum {\alpha ^{2}\beta }{\text{,}}\sum {\alpha \beta \gamma }$$

of the weight $3$.

Of symmetric functions whose representative terms involve two of the $n$ quantities we have the two types in which the repetitional exponents are alike, or different,

$$\textstyle \sum {\alpha ^{i}\beta ^{i}}\equiv \alpha ^{i}\beta ^{i}+\alpha ^{i}\gamma ^{i}+\beta ^{i}\gamma ^{i}+\ldots +\mu ^{i}\nu ^{i}=(i^{2}){\text{,}}$$

$$\textstyle \sum {\alpha ^{i}\beta ^{j}}\equiv \alpha ^{i}\beta ^{j}+\alpha ^{j}\beta ^{i}+\ldots +\mu ^{i}\nu ^{j}+\mu ^{j}\nu ^{i}=(ij){\text{,}}$$

involving ${\scriptstyle {\frac {1}{2}}}n(n-1)$ and $n(n-1)$ terms respectively.

It is now an easy step to the function

$$\textstyle \sum {\alpha _{1}^{i_{1}}\alpha _{2}^{i_{2}}\alpha _{3}^{i_{3}}\ldots \alpha _{s}^{i_{s}}}$$

wherein we have replaced the quantities $\alpha {\text{, }}\beta {\text{, }}\gamma {\text{,}}\ldots \nu$ by the suffixed series $\alpha _{1}{\text{, }}\alpha _{2}{\text{, }}\alpha _{3}{\text{,}}\ldots \alpha _{s}$.

In the partition notation we write the function

$$(i_{1}i_{2}i_{3}\ldots i_{s}){\text{,}}$$

where of course $s$ cannot be greater than $n$.

It involves a number of terms which can be computed when we know the equalities that occur between the numbers $i_{1}{\text{,}}i_{2}{\text{,}}i_{3}{\text{,}}\ldots i_{s}$.

If we are thinking only of numbers, $(i_{1}i_{2}i_{3}\ldots i_{s})$ is a partition of a number $N=i_{1}+i_{2}+i_{3}+\ldots +i_{s}$, and since a partition of $N$ is defined to be any collection of positive integers whose sum is $N$ we may consider the numbers $i_{1}{\text{,}}i_{2}{\text{,}}i_{3}{\text{,}}\ldots i_{s}$ to be in descending order of magnitude. These numbers are called the Parts of the partition and the partition is said to have $s$ parts.

The series of functions denoted by $(i)$ for different integer values of $i$ constitute a first important set. They are sometimes called one-part functions.

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