You'll think it's a well-known expression already, but it's not, at least I couldn't find it. Please read the question.
In how many ways can n identical objects be distributed in r boxes, provided that each box contains a different number of objects, except zero?
I need this information because of the following question.
$∀x_i∈N$ and $x_1<x_2<⋯<x_n$
How many ordered n's are there that satisfy the equation $$x_1+x_2+⋯+x_n = A $$?
For example for A = 11 there are 12 ordered n's which are
$(11), (1,10), (2,9), (3,8), (4,7), (5,6), (1,2,8), (1,3,7), (1,4,6), (2,3,6), (2,4,5), (1,2,3,5)$
