need help with corner orientation formula

[miotatsu]

formula for calculating number of corner orientation cases

okay so
how would you calculate the number of cases to orient 4 corners on U if corners may be affected on D and you use AUF? There are 23 cases but I am looking for the formula you would use to determine that.

 

[TMOY]

If you don't take AUF into account, there are 81 different cases.

3 of them are completely symmetrical (all corners oriented or twisted in the same direction), which gives 3 different cases up to AUF.
6 of them have a diagonal symmetry (UFL and UBR having the same orientation, UFR and UBL also having the same but a different one from the other two), which gives 3 more cases up to AUF.
The remaining 72 have no useful symmetry, which gives 18 more cases up to AUF.

That's 24 cases, 23 if you don't count the skip.

 

[Cubenovice]

U layer: consider all orientations are possible (because the required sum of total twists can be compensated for in the 2nd layer)
3^4 = 81 states - 1 solved is 80
AUF: 80/4 = 20 distinct cases of which 6 are known COLL cases: 14 new algs (of which would affect the D layer)

Edit: Ninja'd