Factorial of a matrix: what could be the use of it? Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma function.
What use might it be to take the factorial of a matrix?  Do any applications come to mind, or does this – for now* – seem to be restricted to the domain of recreational mathematics?
(*Until e.g. theoretical physics turns out to have a use for this, as happened with Calabi–Yau manifolds and superstring theory...)
 A: I could find the following references that take the use of matrix factorial for a concrete applied context:
Coherent Transform, Quantization and Poisson Geometry: In his book  Mikhail Vladimirovich Karasev uses for example the matrix factorial for hypersurfaces and twisted hypergeometric functions.

Artificial Intelligence Algorithms and Applications: In this conference proceeding the Bayesian probability matrix factorial is used in the context of classification of imputation methods for missing traffic data.
Fluid model: Mao, Wang and Tan deal in their paper (Journal of Applied Mathematics and Computing) with a fluid model driven by an $M/M/1$ queue with multiple exponential vacations and $N$-policy.
Construction of coherent states for multi-level quantum systems: In their paper "Vector coherent states with matrix moment problems" Thirulogasanthar and Hohouéto use matrix factorial in context of quantum physics.
Algorithm Optimization: Althought this is a more theoretic field of application, I would like to mention this matrix factorial use case as well. Vladica Andrejić, Alin Bostan and Milos Tatarevic present in their paper improved algorithms for computing the left factorial residues $!p=0!+1!+\ldots+(p−1)!\bmod{p}$. They confirm that there are no socialist primes $p$ with $5<p<2^{40}$. You may take a look into an arXiv version of this paper.
A: The factorial has a straightforward interpretation in terms of automorphisms/permutations as the size of the set of automorphisms.
One possible generalization of matrices is the double category of spans.
So an automorphism $ R ! = R \leftrightarrow R $ over a span $A \leftarrow R \rightarrow B$ ought to be a reasonable generalization.
I usually find it easier to think in terms of profunctors or relations than spans.
The residual/internal hom of profunctors $(R/R)(a, b) = \forall x, R(x, a) \leftrightarrow R(x, b) $ is a kan extension. The kan extension of a functor with itself is the codensity monad. For profunctors and spans the automorphism ought to be a groupoid (a monad in the category of endospans equipped with inverses.)
The factorial is the size of the automorphism group of a set. The automorphism group ought to generalize to a "automorphism groupoid" of a span. I suspect permutation of a matrix ought to be a automorphism groupoid enriched in Vect but this confuses me.
