794] 
589 
794. 
NUMBERS, PARTITION OF. 
[From the Encyclopaedia Britannica, Ninth Edition, vol. xvii. (1884), p. 614.] 
This subject, created by Euler, though relating essentially to positive integer 
numbers, is scarcely regarded as a part of the Theory of Numbers. We consider in 
it a number as made up by the addition of other numbers: thus the partitions of 
the successive numbers 1, 2, 3, 4, 5, 6, &c., are as follows:— 
i; 
2, 11; 
3, 21, 111; 
4, 31, 22, 211, 1111; 
5, 41, 32, 311, 221, 2111, 11111; 
6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111. 
These are formed each from the preceding ones; thus, to form the partitions of 6 we 
take first 6; secondly, 5 prefixed to each of the partitions of 1 (that is, 51); thirdly, 
4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to 
each of the partitions of 3 (that is, 33, 321, 3111); fifthly, 2 prefixed, not to each 
of the partitions of 4, but only to those partitions which begin with a number not 
exceeding 2 (that is, 222, 2211, 21111); and lastly, 1 prefixed to all the partitions of 
5 which begin with a number not exceeding 1 (that is, 111111); and so in other cases. 
The method gives all the partitions of a number, but we may consider different 
classes of partitions: the partitions into a given number of parts, or into not more 
than a given number of parts; or the partitions into given parts, either with 
repetitions or without repetitions, &c. It is possible, for any particular class of parti 
tions, to obtain methods more or less easy for the formation of the partitions either 
of a given number or of the successive numbers 1, 2, 3, «fee. And of course in any 
case, having obtained the partitions, we can count them and so obtain the number 
of partitions.