Conjecturing a mathematical equation to calculate the number of solutions in sudoku squares of any size

We know that based on "n" we can calculate certain interesting values about the sudokus, what I have never been able to understand is why is there not an equation that tells us based on n the total number of solutions that exist?

here is a possible way to do it, or at least a line of research to follow.

In my post "Equations that work for any sudoku of the form n * n" I show some equations that can help us, especially the one for calculating the templates

http://forum.enjoysudoku.com/equations-that-work-for-any-sudoku-of-the-form-n-n-t38848.html

and in my post on "Shidoku Study with Graphs" I show the relationship that exists between Cells, templates and models "

http://forum.enjoysudoku.com/shidoku-study-with-graphs-t38842.html

That said we have:

[Total number of Sudoku Solutions] = [Number of Models] * [Relabeling]

The problem with this formula is that we did not have an equation or a way to determine the [Number of Models]

By representing the models on a graph of the family of hypercubes we can have some clue of how to solve.

Hypercubes are graphs and therefore have some properties, for example, number of vertices, number of edges, number of faces, etc.

The important point is that we have to find the relationship that exists between n and the Number of Models, making use of the knowledge of the value of the Number of Faces of the Hypercube that supports all the Templates of the particular case.

For example, for the 4x4 sudoku that has a value of n = 2 we have that the hypercube that supports its 16 templates in its vertices is the tesseract, and that the number of faces is 24. So we can determine that:

[Number of models] = [Number of Faces of the Tesseract] / [n]

[Number of models] = 24/2

[Number of models] = 12

and therefore the number of solutions is

[Total number of Sudoku Solutions] = [Number of Models] * [Relabeling]

[Total number of Sudoku Solutions] = 12 * 4!

[Total number of Sudoku Solutions] = 12 * 24

[Total number of Sudoku Solutions] = 288

Now for the 9x9 we know that the Number of models is 18,383,222,420,692,992 and that we must work with a Hypercube that has 46,656 Vertices, therefore we would have to know the relationship between the Number of Faces of the Hypercube (46656) and the number of 9x9 models (18,383,222,420,692,992) to see if we can express this relationship in terms of n = 3

We already know from Russell and Jarvis that:

[Total number of Sudoku 9x9 Solutions] = [Number of Models] * [Relabeling]

[Total number of Sudoku 9x9 Solutions] = 18,383,222,420,692,992 * 9!

[Total number of 9x9 Sudoku Solutions] = 18,383,222,420,692,992 * 362880

[Total number of 9x9 Sudoku Solutions] = 6,670,903,752,021,072,936,960

So what I'm needing your help for is so that a real mathematician, who deals with things like the hypercube family and its characteristics, can give me insight into how to calculate the number of faces that exist in hypercubes with 16 vertices, 46,656 vertices and 110,075,314,176 vertices. To see if by generalizing from these three cases we can find the relationship between the value of n, the number of templates, the number of faces of the hypercube that contains them, to finally be able to calculate the number of models that will be generated and in this way also the amount of solutions that these sodokus have.

Best Regards