  Non-commutativity is ubiquitous in mathematical modeling of reality and in
many cases same algebraic structures are implemented in different situations.
Here we consider the canonical commutation relation of quantum theory and
discuss a simple urn model of the latter. It is shown that enumeration of urn
histories provides a faithful realization of the Heisenberg-Weyl algebra.
Drawing on this analogy we demonstrate how the operator normal forms facilitate
counting of histories via generating functions, which in turn yields an
intuitive combinatorial picture of the ordering procedure itself.
