I am studying a set of functions. This seems like something other people would have studied too, and I'd like to know what other people are calling this and how to read up on it.
These "efficiency curve" functions are:
$$f(d_1, d_2, ..., d_n) \in \{0, 1\} $$
where
$$d_i \in \mathbb{N} $$
and
$$f(..., d_i, ...) = 1, d_i > 1 \implies f(..., d_i-1, ...) = 1$$
In two dimensions, this is one-to-one relatable with a monotonic decreasing function.
Examples in two dimensions (green is 1, red is 0):
You can easily count the number of 2d efficiency functions that fit in some boundary ($b_1$, $b_2$). It is:
$$\frac{(b_1 + b_2)!}{b_1! b_2!} $$
Because looking at the function diagonally from the northeast, and distinguishing the tops and the rights of squares, you will always see $d_1$ tops and $d_2$ rights in some order, and every order is a valid and distinct efficiency curve.
In three dimensions you can think of this as placing sugar cubes into a box such that gravity applied down, left, or back does not cause the cubes to move.
You can look at this diagonally too and that picture uniquely defines the curve. Counting those is harder.
Here is an enumeration of all the efficiency functions in three dimensions with boundaries $(2,2,2)$:
I'm interested in studying this problem in $n$ dimensions. Specifically, I'd like to count how many efficiency functions exist in $n$ dimensions with the boundaries $b_1, b_2, ..., b_n$.
But again, I'm not asking you to do my homework (and this is not homework). I'm just asking what other people call this problem and a reference to anybody else that has written about this.