I think I understand that factorial has been generalized using an approach wherein the function(s) evaluate to the normal definition of factorial for integers when applied to an integer but also can be applied to real numbers and even matrixes. What I am wondering is, does such a function continue to have anything to due with permutations so that this generalization yields something like the combinatorial definition for real numbers or is that false?
And if that is false, is there any example of trying to generalize the definition of permutation so that it does apply not just to integers but also real numbers, complex numbers and perhaps even matrixes?
I would also ask if the gamma function when applied to integers yields the number of permutations "by chance" or if there is some interpretation of the gamma function in which it is actually counting permutations?