Séminaire Lotharingien de Combinatoire, B65c (2011), 78 pp.
Pawel Blasiak and Philippe Flajolet
Combinatorial Models of Creation-Annihilation
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
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.
Received: February 28, 2011.
Accepted: June 8, 2011.
Final Version: June 23, 2011.
The following versions are available: