Mathematical theory of partitions. Mathematicians at the University of Florida solved a 30-year-old mathematics problem with the use of the theory of partitions (Explore, Fall 2000). In mathematical terminology, a partition is a representation of an integer as a sum of positive integers.

(For example, the number 3 has three possible partitions:

3, 2 + 1, and 1 + 1 + 1.) The researchers solved the problem by using “colored partitions” of a number, where the colors correspond to the four suits—red hearts, red diamonds, black spades, and black clubs—in a standard 52-card bridge deck. Consider forming colored partitions of an integer.

a. How many colored partitions of the number 3 are possible?

[Hint: One partition is 31; another is 22 + 13.]

b. How many colored partitions of the number 5 are possible?