Constructive covers of a finite set

TL;DR

This paper introduces the concept of constructive k-covers of a finite set, providing a semi-analytic formula involving new integrated Stirling numbers and an optimization method to count such covers exactly.

Contribution

It defines constructive k-covers, introduces integrated Stirling numbers, and develops a summation formula with an optimization procedure to compute their exact count.

Findings

Derived a semi-analytic summation formula for counting constructive k-covers.

Introduced integrated Stirling numbers as a new combinatorial tool.

Developed an optimization-based method to evaluate the formula efficiently.

Abstract

Given positive integers $n,k$ with $k\leq n$, we consider the number of ways of choosing $k$ subsets of $\{1,\ldots,n\}$ in such a way that the union of these subsets gives $\{1,\ldots,n\}$ 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 $\{1,\ldots,n\}$. 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.