Is this series complete, using '0' is not allowed:
$6 = 6$ or $5+1$ or $1+5$ or $4+2$ or $2+4$ or $3+3$ or $4+1+1$ or $1+4+1$ or $1+1+4$ or $1+2+3$ or $1+3+2$ or $2+3+1$ or $2+1+3$ or $3+1+2$ or $3+2+1$ or $1+1+1+3$ or $1+1+3+1$ or $1+3+1+1$ or $3+1+1+1$ or $1+1+1+1+2$ or $1+1+1+2+1$ or $1+1+2+1+1$ or $1+2+1+1+1$ or $2+1+1+1+1$ or $1+1+1+1+1+1$
5 = 5 or 4+1 or 1+4 or 3+2 or 2+3 or 1+1+3 or 3+1+1 or 1+3+1 or 1+2+2 or 2+1+2 or 2+2+1 or 2+1+1+1 or 1+2+1+1 or 1+1+2+1 or 1+1+1+2 or 1+1+1+1+1
4 = 4 or 3+1 or 2+2 or 1+3 or 1+1+2 or 1+2+1 or 2+1+1 or 1+1+1+1
3 = 3 or 1+1+1 or 1+2 or 2+1
2 = 2 or 1+1
1 = 1
The number of partitions in this series is 1,2,4,8,16,25.
Have I missed any, as 25 seems odd to me.
You have missed seven ways. Mainly, $$2+2+2=6$$$$1+1+2+2=6$$$$1+2+1+2=6$$$$1+2+2+1=6$$$$2+1+2+1=6$$$$2+2+1+1=6$$$$2+1+1+2=6$$So there are $32$ ways. However, one thing worth mentioning is that your definition strays from the normal definition of partitions, where two sums that differ only in the order of their summands are considered the same partition. In fact, if the order matters, as in your case, the correct term is a composition.