Intractable enumeration problems are like Russian nesting dolls: structural properties of monomer-dimer coverings on two-dimensional quadratic lattices

Abstract

Counting the number of coverings of $s$ dimers on two-dimensional quadratic lattices is considered as intractable and belongs to \#P-complete class. We reveal the structure of the exact solution to the problem and provide an explicit formula for it, which includes $s-1$ nesting sums. This results in an exponential time complexity of $O(2^s)$. The solution is explicitly determined by a sequence that exhibits double-exponential growth.