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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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