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)