let $$\Pi\binom{n}{k}:=\mathrm{card}\left( \left\lbrace \lbrace \Pi\_1^n\,\cdots\,\Pi\_k^n\rbrace\,|\,0\leq \pi\_{r,c}\in\sum\_{i=1}^k\Pi\_i^n\ni\pi\_{r,c}\leq 1\right\rbrace\right)$$ be the number of sets with $k\leq n$ distinct $n\times n$ permutation matrices *without* common non-zero entry. 



> 
> **Questions:** 
> 
> 
> * has that number already appeared in publications and if yes,
> 	+ what name and/or notation was used?
> * can $\Pi\binom{n}{k}$ be expressed via standard functions of $n$ and $k$?
> * what is the time complexity of calculating $\Pi\binom{n}{k}$?
> 
> 
> 



Calculating the values of a generalization of binomials to permutations
It is a $k\times n$ latin rectangle: write the permutations one per row.


[This paper](https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/on-the-number-of-latin-rectangles/E093CA0EC0A723261F4D635AED40A567) has a nice summary of theoretical and practical methods.


The sum of the permutation matrices can be interpreted as the adjacency matrix of a regular digraph or the biadjacency matrix of a regular bipartite graph. However each digraph/graph can be decomposed into permutation matrices in a varying number of ways, so there is no precise relationship between the counts. 


