 One recovers the function "p"("n") by

One possible generating function for such partitions, taking "k" fixed and "n" variable, is 

More generally, if "T" is a set of positive integers then the number of partitions of "n", all of whose parts belong to "T", has generating function 
This can be used to solve change-making problems (where the set "T" specifies the available coins).As two particular cases, one has that the number of partitions of "n" in which all parts are 1 or 2 (or, equivalently, the number of partitions of "n" into 1 or 2 parts) is
and the number of partitions of "n" in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of "n" into at most three parts) is the nearest integer to ("n" + 3)2 / 12.The asymptotic growth rate for "p"("n") is given by

where formula_19

If "A" is a set of natural numbers, we let "p""A"("n") denote the number of partitions
of "n" into elements of "A".