Geometric juggling with q-analogues

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 steady-state distribution containing $q$-rook polynomial coefficients and $q$-Stirling numbers of the second kind. We show that the equilibrium probabilities of the bounded model can be uniformly approximated with the equilibrium probabilities of a corresponding unbounded model. This observation leads to new limit formulae for $q$-analogues. Keywords: juggling pattern; $q$-Stirling number of the second kind; Ferrers board; Markov process; combinatorial equilibrium