Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/24

where in each term the largest number available is written first, the next largest second, and so on; and in ordering the partitions numbers in descending order of magnitude are in the same relation as are the successive letters of the alphabet in dictionary order. The alternative method is to adopt the numerical order subject to the rule that the partitions are to be arranged in ascending order in regard to the number of parts involved.

This would give the expression $h_{6}$ $=(6)+(51)+(42)+(33)+(411)+(321)+(222)\quad$ $\quad \quad \;+(3111)+(2211)+(21111)+(111111)$.

When we have to write out the compositions associated with a given partition we adopt numerical order.

Thus associated with the partition $(321)$ of the number 6 we have the compositions $(321),(312),(231),(213),(132),(123)$.