  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.
