Abstract
I propose to discuss the following problem: “To find in how many ways a number
q
can be made up of the elements
a, b, c,..
each element being repeatable an indefinite number of times.” The required number of partitions is represented by the notation P
(a, b, c, ..)q
, and we have, as is well known, P
(a, b, c, ..)q
= coefficient in
x
q
in 1/ (1-
x
a
) (1-
x
b
) (1-
x
c
) ..' where the expansion is to be effected in ascending powers of
x
. It may be as well to remark that each element is to be considered as a separate and distinct element, notwithstanding any equalities which may exist between the numbers
a, b, c,
..; thus, although
a=b
, yet
a + a + a +
&c. and
a + a + b +
&c. are to be considered as two different partitions of the number, and so in all similar cases. The solution of the problem is thus seen to depend upon the theory, to which I now proceed, of the expansion of algebraical fractions.
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