  Quantum physics has revealed many interesting formal properties associated
with the algebra of two operators, A and B, satisfying the partial commutation
relation AB-BA=1. This study surveys the relationships between classical
combinatorial structures and the reduction to normal form of operator
polynomials in such an algebra. The connection is achieved through suitable
labelled graphs, or "diagrams", that are composed of elementary "gates". In
this way, many normal form evaluations can be systematically obtained, thanks
to models that involve set partitions, permutations, increasing trees, as well
as weighted lattice paths. Extensions to q-analogues, multivariate frameworks,
and urn models are also briefly discussed.
