What is the partition function?

The partition function, p(n), counts the number of ways the integer n can be written as a sum of positive integers. For example, p(4)=5 because there are 5 way to write the number 4 as a sum of whole numbers:


The partition function grows rapidly. With some hard work one could check that p(10)=42. It is a fact, which one could not verify by hand, that p(100) = 190,569,292 and p(1000)=24,061,467,864,032,622,473,692,149,727,991.