Let n be a positive integer. A partition of n is a way of writing n as a sum of positive integers. For example, all the partitions of 5 are:

   1+1+1+1+1
   1+1+1+2
   1+1+3
   1+2+2
   1+4
   2+3
   5

Only three out of the seven partitions of 5 correspond to no repetitions:

   1+4
   2+3
   5

Four out of the seven patitions of 5 contain odd numbers of summands:

   1+1+1+1+1
   1+1+3
   1+2+2
   5