number of ordered partitions of integer How to evaluate the number of ordered partitions of the positive integer $ 5 $?
Thanks!

Counting in binary the groups of 1s or 0s form the partitions. Half are the same so there are 2^(n-1). As to be expected this gives the same results as the gaps method, but in a different order.
Groups
0000 4 
0001 3,1 
0010 2,1,1 
0011 2,2 
0100 1,1,2 
0101 1,1,1,1 
0110 1,2,1 
0111 1,3

Gaps
000 4
001 3,1
010 2,2
011 2,1,1
100 1,3
101 1,2,1
110 1,1,2
111 1,1,1,1

