and generally if each part may occur at most k - 1 times it is
| 1 - ζkxka | · | 1 - ζkxkb | · | 1 - ζkxkc | · ... |
| 1 - ζxa | 1 - ζxb | 1 - ζxc |
It is thus easy to form generating functions for the partitions of numbers into parts subject to various restrictions. If there be no restriction in regard to the numbers of the parts, the generating function is
| 1 |
| 1 - xa. 1 - xb. 1 - xc. ... |
and the problems of finding the partitions of a number n, and of determining their number, are the same as those of solving and enumerating the solutions of the indeterminate equation in positive integers
ax + by + cz + ... = n.
Euler considered also the question of enumerating the solutions of the indeterminate simultaneous equation in positive integers
| ax + by + cz + ... = n a′x + b′y + c′z + ... = n′ a″x + b″y + c″z + ... = n″ |
which was called by him and those of his time the “Problem of the Virgins.” The enumeration is given by the coefficient of xnyn′zn″ ... in the expansion of the fraction
| 1 |
| (1 - xaybzc ...) (1 - xa′yb′zc′ ...) (1 - xa″yb″zc″ ...) ... |