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Geometric juggling with $$q$$-analogues. (English) Zbl 1309.05028
Summary: We derive a combinatorial equilibrium for bounded juggling patterns with a random, $$q$$-geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a stationary distribution containing $$q$$-rook polynomial coefficients and $$q$$-Stirling numbers of the second kind. We show that the stationary probabilities of the bounded model can be uniformly approximated with the stationary probabilities of a corresponding unbounded model. This observation leads to new limit formulae for $$q$$-analogues.

##### MSC:
 05A30 $$q$$-calculus and related topics 05A15 Exact enumeration problems, generating functions 11B73 Bell and Stirling numbers
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