Mathematics Problem Of the Week

Fall 2001

POW #8

Products of Partitions

A partition of a positive integer is a way of writing the integer as a sum of positive integers.

For example, 8=5+3, 8=2+2+1+1+1+1 and 8=8 are all partitions of 8. The order doesn't matter, although it is traditional to write the terms in descending order.

Once we have a partition of N, we can look at the product of all of the terms in the partition. The partitions of 8 shown above would have products 5x3=15, 2x2x1x1x1x1=4 and 8=8. We are interested in the partition(s) that produce largest product.

(a) Out of all possible partitions of 8, which give(s) the largest product?

(b) More generally, for a positive integer N, which partition(s) of N will give the largest product? Does the answer depend on any features of N? Justify your answers.

Due Friday, November 9^th at Noon.