Constructive covers of a finite set

Given positive integers n,k with k≤ n, we consider the number of ways of choosing k subsets of {1,…,n} in such a way that the union of these subsets gives {1,…,n} and they are not subsets of each other. We refer to such choices of sets as constructive k-covers and provide a semi-analytic summation formula to calculate the exact number of constructive k-covers of {1,…,n}. Each term in the summation is the product of a new variant of Stirling numbers of the second kind, referred to as integrated Stirling numbers, and the cardinality of a certain set which we calculate by an optimization-based procedure with no-good cuts for binary variables.