A partition of a positive integer n is a list of positive integers a1.a2...., ak such that a1 + a2 +•••+ ak = n. For example, the following are distinct partitions of 5.

5 1. 1.1. 2 1.2.2 1. 1.1. 1.1

The order of the list doesn't matter: 1. 2.2 is the same partition as 2. 1.2. There is a natural partial ordering on the set of partitions of n: if P1 and P2 are partitions. define P1 s P2 if P1 can be obtained by combining parts of P2.

For example. 1. 2. 2 £ 1. 1. 1. 1.1 because 1. 2. 2 = 1.1 + 1. 1 + 1. On the other hand. 2.3 and 1. 4 are incomparable elements in this poset.

(a) Write the partitions of 6 in a Hasse diagram. (There are 11 partitions of 6.)

(b) Is this a total ordering? Why or why not?