590 
NUMBERS. 
[794 
Another method is by Arbogast’s rule of the last and the last but one; in fact, 
taking the value of a to be unity, and, understanding this letter in each term, the 
rule gives b; c, b 2 ; d, be, b 3 ; e, bd, c 2 , b 2 c, b 4 , &c., which, if b, c, d, e, &c., denote 
1, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively. 
An important notion is that of conjugate partitions. Thus a partition of 6 is 42; 
writing this in the form , and summing the columns instead of the lines, we 
obtain the conjugate partition 2211; evidently, starting from 2211, the conjugate 
partition is 42. If we form all the partitions of 6 into not more than three parts, 
these are 
6, 51, 42, 33, 411, 321, 222, 
and the conjugates are 
111111, 21111, 2211, 222, 3111, 321, 33, 
where no part is greater than 3; and so, in general, we have the theorem, the number 
of partitions of n into not more than k parts is equal to the number of partitions 
of n with no part greater than k. 
We have for the number of partitions an analytical theory depending on generating 
functions; thus for the partitions of a number n with the parts 1, 2, 3, 4, 5, &c., 
without repetitions, writing down the product 
1 + ¿c. 1 + # 2 .1 -f- # 3 . 1 + ¿c 4 ..., =1 -f x + x 2 + 2x 3 + ... + Nx n + ..., 
it is clear that, if x a , x/, xv, ... are terms of the series x, x 2 , x 3 , ... for which 
a. + (3 + y + ... =n, then we have in the development of the product a term x n , and 
hence that, in the term Nx n of the product, the coefficient N is equal to the number 
of partitions of n with the parts 1, 2, 3,..., without repetitions; or say that the 
product is the generating function (G. F.) for the number of such partitions. And so 
in other cases we obtain a generating function. 
Thus for the function 
^ ^ ^— > = 1 + x -f 2x 2 + ... + Nx n + ..., 
1— X.l— X 2 .1— X? ... 
observing that any factor 1/1 — x l is =1 +x l + x 2l +..., we see that, in the term Nx n , 
the coefficient is equal to the number of partitions of n, with the parts 1, 2, 3, ..., 
with repetitions. 
Introducing another letter z, and considering the function 
1 + xz. 1 4-x 2 z . 1 + a?z ..., =1 +z(x + x 2 +...) + ...+ Nx n z k + ..., 
we see that, in the term Nx n z k of the development, the coefficient N is equal to the 
number of partitions of n into k parts, with the parts 1, 2, 3, 4, ..., without repetitions. 
And similarly, considering the function 
1 
1 — xz. 1 — X 2 Z . 1 — X 3 Z ... 
, —l+z{x+ x 2 +...)+...+ Nx n z k + ..., 
we see that, in the term Nx 1l z k of the development, the coefficient N is equal to the 
number of partitions of n into k parts, with the parts 1, 2, 3, 4,..., with repetitions.