Number of ways to choose $k$ permutations of $n$ so that any two are different in every position

Question

I'd like to calculate the number of ways to select $k$ permutations of $1...n$ so that any two permutations are different in every position.

Formally, choosing permutations $p_1, p_2, ... p_k$, where $\{p_{i,1}, p_{i,2} ... p_{i,n}\}=\{1,2...n-1,n\}$, for every $1 \leq i,j \leq k,~i \neq j,~1 \leq s \leq n~~~p_{i,s} \neq p_{j,s}$ is satisfied.

Any ideas, formulas, algorithms are welcome.

Answer 1

Echoing @PM 2Ring's comments, this problem is the same as counting the number of latin rectangles with width $n$ and height $k$. A reasonably thorough presentation is given by Douglas S. Stones of Monash University: http://www.combinatorics.org/ojs/index.php/eljc/article/download/v17i1a1/pdf