Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/23

A monomial symmetric function, as defined in Art. 5, is the sum of a number of different combinations of the same type. In writing out at length these combinations of quantities $\alpha ,\beta ,\ldots \nu$ we may adopt what is called the ‘dictionary’ (or alternatively ‘alphabetical’) order.

In any particular combination we write the $\alpha$'s first, then the $\beta$'s, and so forth; also one combination is given priority of another if, considering the two combinations to be words, the dictionary would give the one word before the other.

This allusion to the dictionary, with which all persons are familiar, seems to define shortly and clearly the principle of order usually adopted. Thus we write the monomial function of four quantities $\alpha ,\beta ,\gamma ,\delta$ $\textstyle {\sum }\alpha ^{2}\!\beta \gamma$ in the order $\;\alpha ^{2}\!\beta \gamma +\alpha ^{2}\!\beta \delta +\alpha ^{2}\!\gamma \delta +\alpha \beta ^{2}\!\gamma +\alpha \beta ^{2}\!\delta +\alpha \beta \gamma ^{2}\!$ $+\alpha \beta \delta ^{2}\!+\alpha \gamma ^{2}\!\delta +\alpha \gamma \delta ^{2}\!+\beta ^{2}\!\gamma \delta +\beta \gamma ^{2}\!\delta +\beta \gamma \delta ^{2}\!$, the dictionary order being in evidence both in each combination and in the order of the combinations.

Another order is sometimes useful. We may have, in each combination, the repetitional numbers always in the same order as they appear in the representative combination but subject to this rule, the letters in dictionary order.

The combinations thus written would then be arranged in dictionary order. Thus we might write $\textstyle {\sum }\alpha ^{2}\!\beta \gamma$ $=\alpha ^{2}\!\beta \gamma +\alpha ^{2}\!\beta \delta +\alpha ^{2}\!\gamma \delta +\beta ^{2}\!\alpha \gamma +\beta ^{2}\!\alpha \delta +\beta ^{2}\!\gamma \delta$ $+\gamma ^{2}\!\alpha \beta +\gamma ^{2}\!\alpha \delta +\gamma ^{2}\!\beta \delta +\delta ^{2}\!\alpha \beta +\delta ^{2}\!\alpha \gamma +\delta ^{2}\!\beta \gamma$.

12. Again, frequently we have to write out at length the permutations of a given combination of letters. Here again it is usual to adopt the dictionary order, each permutation being regarded as a dictionary word. Thus the twelve permutations of $\alpha \alpha \beta \gamma$ are written $\alpha \alpha \beta \gamma \quad \alpha \alpha \gamma \beta \quad \alpha \beta \alpha \gamma \quad \alpha \beta \gamma \alpha \quad \alpha \gamma \alpha \beta \quad \alpha \gamma \beta \alpha$ $\beta \alpha \alpha \gamma \quad \beta \alpha \gamma \alpha \quad \beta \gamma \alpha \alpha \quad \gamma \alpha \alpha \beta \quad \gamma \alpha \beta \alpha \quad \gamma \beta \alpha \alpha$.

13. When we have to write out symmetric functions, of the same weight, expressed in partition notation, we usually adopt numerical order, the meaning of which will be clear from the example

$h_{6}$ $=(6)+(51)+(42)+(411)+(33)+(321)+(3111)\quad$ $\quad \quad \;+(222)+(2211)+(21111)+(111111)$,