Combinatorially interpreting generalized Stirling numbers

The Stirling numbers of the second kind n k (counting the number of partitions of a set of size n into k non-empty classes) satisfy the relation ( x D ) n f ( x ) = ? k ? 0 n k x k D k f ( x ) where f is an arbitrary function and D is differentiation with respect to x . More generally, for every word w in alphabet { x , D } the identity w f ( x ) = x ( # ( x 's?in? w ) - # ( D 's?in? w ) ) ? k ? 0 S w ( k ) x k D k f ( x ) defines a sequence ( S w ( k ) ) k of Stirling numbers (of the second kind) of w . Explicit expressions for, and identities satisfied by, the S w ( k ) have been obtained by numerous authors, and combinatorial interpretations have been presented.Here we provide a new combinatorial interpretation that, unlike previous ones, retains the spirit of the familiar interpretation of n k as a count of partitions. Specifically, we associate to each w a quasi-threshold graph G w , and we show that S w ( k ) enumerates partitions of the vertex set of G w into classes that do not span an edge of G w . We use our interpretation to re-derive a known explicit expression for S w ( k ) , and in the case w = ( x s D s ) n to find a new summation formula linking S w ( k ) to ordinary Stirling numbers. We also explore a natural q -analog of our interpretation.In the case w = ( x r D ) n it is known that S w ( k ) counts increasing, n -vertex, k -component r -ary forests. Motivated by our combinatorial interpretation we exhibit bijections between increasing r -ary forests and certain classes of restricted partitions.

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

[2]  Bruce E. Sagan,et al.  A Chromatic Symmetric Function in Noncommuting Variables , 2001 .

[3]  Leonard Carlitz,et al.  $q$-Bernoulli numbers and polynomials , 1948 .

[4]  Toufik Mansour,et al.  Representing elements of the Weyl algebra by labeled trees , 2013 .

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

[6]  France,et al.  Heisenberg–Weyl algebra revisited: combinatorics of words and paths , 2008, 0904.1506.

[7]  Emil Grosswald,et al.  The Theory of Partitions , 1984 .

[8]  Kraków,et al.  Combinatorics and Boson normal ordering: A gentle introduction , 2007, 0704.3116.

[9]  France,et al.  Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem , 2005 .

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

[11]  Waleed A. Al-Salam,et al.  q‐Bernoulli numbers and polynomials , 1958 .

[12]  Metin Arik,et al.  Hilbert spaces of analytic functions and generalized coherent states , 1976 .

[13]  Philippe Flajolet,et al.  COMBINATORIAL MODELS OF CREATION{ANNIHILATION , 2010, 1010.0354.

[14]  G. Birkhoff.  A Determinant Formula for the Number of Ways of Coloring a Map , 1912 .

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

[16]  Toufik Mansour,et al.  Normal ordering problem and the extensions of the Stirling grammar , 2013, 1308.0169.

[17]  A. M. Navon,et al.  Combinatorics and fermion algebra , 1973 .

[18]  Pietro Codara,et al.  A simple combinatorial interpretation of certain generalized Bell and Stirling numbers , 2013, Discret. Math..

[19]  Leonard Carlitz,et al.  On Arrays of Numbers , 1932 .

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  JAY R. GOLDMAN,et al.  Rook theory III. Rook polynomials and the chromatic structure of graphs , 1978, J. Comb. Theory, Ser. B.

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

[23]  Leonard Carlitz.  On A Class of Finite Sums , 1930 .

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

[25]  Martin Griffiths.  Sitting Down for Dinner at a Twin Convention , 2013 .

[26]  Gerard J. Chang,et al.  Quasi-threshold Graphs , 1996, Discret. Appl. Math..

[27]  Wolfdieter Lang,et al.  Combinatorial Interpretation of Generalized Stirling Numbers , 2009 .