Intersecting generalised permutations

For any positive integers $k,r,n$ with $r\le \min \{k,n\}$, let ${\mathcal{P}}_{k,r,n}$ be the family of all sets ${(x_1,y_1), \dots, (x_r,y_r)}$suchthat$x_1, \dots, x_r$aredistinctelementsof$[k] = {1, \dots, k}$and$y_1, \dots, y_r$aredistinctelementsof$[n]$\mathrm{.}Thefamilies$$\mathcal{P}${n,n,n}$and$$\mathcal{P}${n,r,n}$describepermutationsof$[n]$and$r$-partialpermutationsof$[n]$,respectively\mathrm{.}If$k \leq n$,then$$\mathcal{P}${k,k,n}$describespermutationsof$k$-elementsubsetsof$[n]$. A family $\mathcal{A}$ of sets is said to be intersecting if every two members of $\mathcal{A}$ intersect. In this note we use Katona’s elegant cycle method to show that a number of important Erd\H{o}s-Ko-Rado-type results by various authors generalise as follows: the size of any intersecting subfamily $\mathcal{A}$ of ${\mathcal{P}}_{k,r,n}$ is at most ${k-1 \choose r-1}\frac{(n-1)!}{(n-r)!}$,andtheboundisattainedifandonlyif$$\mathcal{A}$ = {A \in $\mathcal{P}${k,r,n} \colon (a,b) \in A}$forsome$a \in [k]$and$b \in [n]$.