One partition of 8 is 5 + 3, but if we then partition each of the 5 and 3 we could get (3+2) + (2+1), and then partition again to get ((2+1)+(1+1)) + ((1+1)+1) and finally (((1+1)+1)+(1+1)) + ((1+1)+1). 5+3 could also be expanded as (4+1)+(2+1), then ((2+2)+1)+((1+1)+1), then (((1+1)+(1+1))+1)+((1+1)+1).
This question is about viewing "+" as a binary operation , so 1+1+1+1 would have to written as either (1+1)+(1+1) or ((1+1)+1)+1.
Every partition can be written as a such nested partition of 1s. It is still order independent, but associative dependent.
For a given number $n$, how many associative dependent binary-operation nested partitions of 1s are there of $n$ ?
What is known about this function ?