Literature is rife with results on classical derangements, that is, permutations on a set of elements that leave no element fixed. Generalizations also abound. In 'Matchings, Derangements, Rencontres', we have a pretty good description of the numbers $D(n,k,r)$ of $k$-permutations on $n$ elements that leave all but $r$ elements fixed. $q$- and even $p,q$- analogues of derangement numbers have been studied.
I am interested in another generalization of derangements. A classical derangement leaves no element of one given arrangement fixed. Given $k$ arrangements $\sigma_i$ $(1 \leq i \leq k)$, of the same $n$ elements, a $k$-derangement is a permutation $\rho$ (of the $n$ elements) that at no point coincides with any $\sigma_i$. In particular, a $1$-derangement is a classical derangement.
The only source I could find discussing this topic is this one. While certainly enlightening, I would like to know how to actually, theoretically, calculate the number $D(\sigma_1, \dots, \sigma_k)$ of $k$-derangements. Any ideas, or references?