Path decompositions of digraphs and their applications to Weyl algebra

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra, similar as Eulerian tours applicable for Amitsur–Levitzki theorem. We introduce the $G$-Stirling functions which enumerate decompositions by sources (and sinks) of paths.