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A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

Abstract : 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.
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Miguel Méndez, Adolfo Rodríguez. A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.557-570. ⟨hal-01185141⟩

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