Climbing/Descending the Multidimensional Integer Ladder

Question

This is basically a follow-up to Climbing/Descending the Integer Ladder, but in multiple dimensions. It's basically just an index counting problem, but combinatoric blow-up makes it interesting.

In that question, we had a single ladder that we could climb up or down, and we wanted to hit some target integer $k$ over $m$ steps. I'd like to extend this to higher-dimensions.

Let's assume we have a sequence of numbers of steps, $\{m_i\}$, where $i=1..N$ (i.e. this is an $N$-dimensional problem) and each of these $\{m_i\}$ can be partitioned over $N$ ladders, e.g. if $N=7$ and $m_3=5$ then I can do a single step on 5 of the 7 ladders, or 2 on the first, and 1 on three others, etc.

The question is then, how many different ways can we do this partitioning so that after we're done with all $N$ climbing/descending operations we end up at $0$ on all of the ladders. All of the ladders are equivalent, so the following paths are equivalent in this kind of picture (for $\{m_i\}=\{3, 3, 2\}$)

{{-3, 0, 0}, {2, -1,0}, {1, 1,0}} <-> {{0, -3, 0}, {-1, 2, 0}, {1, 1, 0}}

since all we did was transpose the indices. In general, I actually have that some of the ladders will be equivalent and some won't (i.e. there'll be a prescribed transposition symmetry to the indices), but I think that'll make the solutions more complicated and the problem less fun.

Here's the naive approach that takes no advantage of symmetry and is just based on pruning the total list of candidates.

ladderUpAndDownPaths[changes : {__Integer}] :=
 Block[
  {
   maxDim = Max[changes],
   displacements
   },
  displacements =
   Function[{dim},
     Select[
      Tuples[Range[-dim, dim], maxDim],
      Total[Abs[#]] == dim &
      ]
     ] /@ changes;
  Select[Tuples[displacements], AllTrue[Total[#], EqualTo[0]] &]
  ]

which gives

ladderUpAndDownPaths[{2, 2}]//Column

{{-2,0},{2,0}}
{{-1,-1},{1,1}}
{{-1,1},{1,-1}}
{{0,-2},{0,2}}
{{0,2},{0,-2}}
{{1,-1},{-1,1}}
{{1,1},{-1,-1}}
{{2,0},{-2,0}}

but of course this blows up

ladderUpAndDownPaths[{3, 3, 2}]

{{-3,0,0},{2,-1,0},{1,1,0}}
{{-3,0,0},{2,0,-1},{1,0,1}}
{{-3,0,0},{2,0,1},{1,0,-1}}
{{-3,0,0},{2,1,0},{1,-1,0}}
{{-2,-1,0},{1,1,-1},{1,0,1}}
{{-2,-1,0},{1,1,1},{1,0,-1}}
{{-2,-1,0},{1,2,0},{1,-1,0}}
{{-2,-1,0},{2,-1,0},{0,2,0}}
{{-2,-1,0},{2,0,-1},{0,1,1}}
{{-2,-1,0},{2,0,1},{0,1,-1}}
{{-2,-1,0},{3,0,0},{-1,1,0}}
{{-2,0,-1},{1,-1,1},{1,1,0}}
{{-2,0,-1},{1,0,2},{1,0,-1}}
{{-2,0,-1},{1,1,1},{1,-1,0}}
{{-2,0,-1},{2,-1,0},{0,1,1}}
{{-2,0,-1},{2,0,-1},{0,0,2}}
...
{{2,0,1},{-2,1,0},{0,-1,-1}}
{{2,0,1},{-1,-1,-1},{-1,1,0}}
{{2,0,1},{-1,0,-2},{-1,0,1}}
{{2,0,1},{-1,1,-1},{-1,-1,0}}
{{2,1,0},{-3,0,0},{1,-1,0}}
{{2,1,0},{-2,0,-1},{0,-1,1}}
{{2,1,0},{-2,0,1},{0,-1,-1}}
{{2,1,0},{-2,1,0},{0,-2,0}}
{{2,1,0},{-1,-2,0},{-1,1,0}}
{{2,1,0},{-1,-1,-1},{-1,0,1}}
{{2,1,0},{-1,-1,1},{-1,0,-1}}
{{3,0,0},{-2,-1,0},{-1,1,0}}
{{3,0,0},{-2,0,-1},{-1,0,1}}
{{3,0,0},{-2,0,1},{-1,0,-1}}
{{3,0,0},{-2,1,0},{-1,-1,0}}
```