Dividing Permutations in the Semiring of Functional Digraphs

Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a semiring with an interesting multiplicative structure. For instance, we do not know if the following division problem can be solved in polynomial time: given two functional digraphs A and B, does A divide B? That A divides B means that there exist a functional digraph X such that AX is isomorphic to B, and many such X can exist. We can thus ask for the number of solutions X. In this paper, we focus on the case where B is a permutation, that is, a disjoint union of cycles. There is then a naïve sub-exponential algorithm to compute the number of non-isomorphic solutions X, and our main result is a polynomial algorithm when A is fixed. It uses a divide-and-conquer technique that should be useful for further developments on the division problem.