Counting polyominoes on (hyper-)cubes

Question

This challenge like some of my previous challenges will have you counting free polyforms, which are generalizations of Tetris pieces.

This code-golf challenge will have you count polyomino-like polyforms on hypercubes. In particular, this challenge is to write a program that takes in three parameters:

• n, which represents an \$n\$-dimensional hypercube,
• m, which represents \$m\$-dimensional faces of the hypercube, and
• k, which represents the number of cells in the polyform,

and outputs the number of ways to choose \$k\$ (\$m\$-dimensional) faces on the \$n\$-cube such that the \$m\$-faces are connected at \$(m-1)\$-faces. These polyforms are "free" which means they should be counted up to the rotations/reflections of the \$n\$-cube.

Again, this is a code-golf challenge, so shortest code wins.

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Example 1

Okay, this is all very abstract, so this warrants an example.

When n=3, we're talking about the \$3\$-dimensional (ordinary) cube. When m=2 this means we're talking about the \$2\$-dimensional (square) faces. And we're talking about k of these, joined along \$1\$-dimensional faces (edges).

When k=3, there are two such polyforms (on the left) up to rotations/reflections of the cube. When k=4 there are also two polyforms (on the right).

Example 2

In this second example, n=3 still, so we're again talking about the \$3\$-dimensional (ordinary) cube. When m=1 this means we're talking about the \$1\$-dimensional faces (edges). And we're talking about k of these, joined along \$0\$-dimensional faces (corners).

When k=4 there are four such polyforms.

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Data

n | m | k | f(n,m,k)
--+---+---+---------
3 | 2 | 3 | 2       (Example 1, left)
3 | 2 | 4 | 2       (Example 1, right)
3 | 1 | 4 | 4       (Example 2)
2 | 1 | 2 | 1
3 | 0 | 0 | 1
3 | 0 | 1 | 1
3 | 0 | 2 | 0
3 | 1 | 3 | 3
3 | 1 | 5 | 9 
3 | 1 | 6 | 14
3 | 1 | 7 | 19
3 | 1 | 8 | 16
3 | 1 | 9 | 9
3 | 3 | 0 | 1
3 | 3 | 1 | 1
3 | 3 | 2 | 0
4 | 1 | 4 | 7
4 | 1 | 5 | 21
4 | 1 | 6 | 72
4 | 1 | 7 | 269
4 | 1 | 8 | 994
4 | 1 | 9 | 3615
4 | 2 | 3 | 5
4 | 2 | 4 | 12
4 | 2 | 5 | 47
5 | 1 | 4 | 7
5 | 1 | 5 | 27
5 | 2 | 0 | 1
5 | 2 | 1 | 1
5 | 2 | 2 | 1
5 | 2 | 3 | 5
5 | 2 | 4 | 20
5 | 3 | 4 | 16
5 | 3 | 5 | 73
5 | 4 | 4 | 3
6 | 1 | 6 | 121

Answer 1

Python 3, 389

import itertools as I
S=sorted
P=I.product
def C(n,m,k):
 Q=[((-1,)*(n-m)+(0,)*m,)]
 for i in' '*(k-1):Q=set(tuple(S(q+(v,)))for q in Q for v in P(*[(-1,0,1)]*n)if sum(map(abs,v))==n-m if not v in q and any(sum((a!=b)*(1+2*a*b)for a,b in zip(v,u))==2for u in q))
 return sum(all(S(q)<=S(zip(*r))for X in I.permutations(zip(*q))for r in P(*((p,tuple(-x for x in p)) for p in X)))for q in Q)

Try it online!

Basically just finds all connected polyominos, and discards ones which can be rotated into a lexicographically smaller polyomino, with rotations being brute-forced.

Can definitely be improved but it's my bedtime.