Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

We consider the numbers arising in the problem of normal ordering of expressions in boson creation a† and annihilation a operators ([a,a†]=1). We treat a general form of a boson string (a†)rnasn...(a†)r2as2(a†)r1as1 which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobinski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem