Following the advice @whuber of describing the combinatoric interpretation of $\kappa^{(k)}_{\alpha-\beta}$, I figured out the answer.
Those counts appear in the powers of mapping of subsets of a range $[0, D]$. You define $K^{(\alpha\beta)}$ with $\alpha > \beta$ as the function taking a pair of sets of size $\alpha$ and $\beta$ respectively and returning 1 when they intersect and 0 otherwise. Then you arrange all subsets of $[0, D]$ in a basis set so that you define $K$ as the matrix mapping all the subsets together (it is a triangular matrix). It is composed of many $K^{(\alpha\beta)}$ blocks. Those coefficient $\kappa^{(k)}_{\alpha-\beta}$ appear when you take the $k^{th}$ power of $K$. More precisely, using a kinda of notation shortcut $$ K = \sum_{\alpha = 2}^D\sum_{\beta=1}^{\alpha-1} K^{(\alpha\beta)} $$ and $$ K^k = \sum_{\alpha = k+1}^D\sum_{\beta=1}^{\alpha-k} \kappa^{(k)}_{\alpha-\beta} K^{(\alpha\beta)} $$
For two subsets $S, S'$ of the range $[0, D]$ with $S \subset S'$, $K^k(S, S')$ counts the number of ways you can split $S'\setminus S$ in $k$ subsets exactly. This corresponds to the Stirling numbers, $S(n, k)$, with an extra factorial for the ordering of the partition selection $$ \kappa^{(k)}_{\alpha-\beta} = k! S(n, k) $$
It corresponds to the definition of Triangle numbers, and are similar to the the Worpitzky numbers.