bounded partitions and bounded signed partitions of integers

Question

Define a bounded signed partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation:

$$n = \pm a_{1} \pm a_{2} \pm a_{3} \pm \dots \pm a_{m}$$ where each $a_{i}$ is a integer in the range $[0,h]$

Define a bounded partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation:

$$n = a_{1} + a_{2} + a_{3} + \dots + a_{m}$$ where each $a_{i}$ is a integer in the range $[0,h]$

Are there relations for the number of both partitions of $n$ in terms of $m$, $h$ and $n$?

Are there strong asymptotic forms?

In the plain bounded partition case what is the asymptotic if each $a_{i} < h_{i}$ and $h_{i} = h_{i+1} - 1$? (ordering on the maximum sizes and not on the summands)

Answer 1

Since you include 0 and do not ask that the a_i are increasing with i, your questions are morally equivalent to counting lattice points in the m dimensional plane $n= \sum x_i,$ except you want those inside a certain m-cube of side length h or 2h, depending on when sign matters. This suggests the correspondence between the two problems of adding h to all coordinates in a signed partition to get an unsigned partition for n+ mh with values in 0 to 2h.

For the unsigned portion and for n at most h, the answer is a simple combinatorial arrangment giving (n+m-1) choose (m-1) such partitions, and the same value serves as a weak upper bound when n> h. Although you can check the literature for this, be mindful that most references will assume an ordered partition, meaning $a_1 \leq a_2 \leq \ldots$ .