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.

[1]  P. Blasiak.  Combinatorics of boson normal ordering and some applications , 2005 .

[2]  Anna Varvak.  Rook numbers and the normal ordering problem , 2005, J. Comb. Theory, Ser. A.

[3]  UK,et al.  The general boson normal ordering problem , 2003 .

[4]  Kraków,et al.  Some useful combinatorial formulas for bosonic operators , 2004, quant-ph/0405103.

[5]  UK,et al.  Boson normal ordering via substitutions and Sheffer-type polynomials , 2005 .

[6]  On Generalizations of the Stirling Number Triangles 1 , 2000 .

[7]  W. Louisell.  Quantum Statistical Properties of Radiation , 1973 .

[8]  James Dolan,et al.  From Finite Sets to Feynman Diagrams , 2001 .

[9]  Wolfdieter Lang,et al.  On Generalizations of the Stirling Number Triangles , 2000 .

[10]  Jacob Katriel,et al.  Bell numbers and coherent states , 2000 .

[11]  M. M. Miller,et al.  Fundamentals of Quantum Optics , 1968 .

[12]  A. Joyal.  Une théorie combinatoire des séries formelles , 1981 .

[13]  Walter Murray,et al.  Sequential Quadratic Programming Methods for Large-Scale Problems , 1997, Comput. Optim. Appl..

[14]  Matthias Schork,et al.  On the combinatorics of normal ordering bosonic operators and deformations of it , 2003 .

[15]  P. Roman.  Introduction to quantum field theory , 1969 .

[16]  Jacob Katriel,et al.  Combinatorial aspects of boson algebra , 1974 .

[17]  UK,et al.  Normal Order:. Combinatorial Graphs , 2004 .

[18]  Philippe Flajolet,et al.  ANALYTIC COMBINATORICS — SYMBOLIC COMBINATORICS , 2002 .

[19]  T. Mansour,et al.  Involutions avoiding the class of permutations in Sk with prefix 12 , 2007 .

[20]  A. I. Solomon,et al.  Deformed Bosons: Combinatorics of Normal Ordering , 2004, quant-ph/0410226.

[21]  Jacob Katriel,et al.  Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers , 1992 .

[22]  Richard Ehrenborg,et al.  Juggling and applications to q-analogues , 1996, Discret. Math..

[23]  R. J. Crewther,et al.  Introduction to quantum field theory , 1995, hep-th/9505152.

[24]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[25]  A. I. Solomon,et al.  The Boson Normal Ordering Problem and Generalized Bell Numbers , 2002 .

[26]  S. Berman,et al.  Nuovo Cimento , 1983 .