Geodesic
A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

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We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers. 
@article{DMTCS_2008_special_255_a15,
     author = {M\'endez, Miguel and Rodr{\'\i}guez, Adolfo},
     title = {A {Combinatorial} {Model} for $q${-Generalized} {Stirling} and {Bell} {Numbers}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
     year = {2008},
     doi = {10.46298/dmtcs.3607},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3607/}
}
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%A Rodríguez, Adolfo
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%J Discrete mathematics & theoretical computer science
%D 2008
%V DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
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Méndez, Miguel; Rodríguez, Adolfo. A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi : 10.46298/dmtcs.3607. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3607/

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