Geometric juggling with qq-analogues

We derive a combinatorial equilibrium for bounded juggling patterns with a random, qq-geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a stationary distribution containing qq-rook polynomial coefficients and qq-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 qq-analogues.