Geometric juggling with q-analogues
We derive a combinatorial equilibrium for bounded juggling patterns with a random, q-geometric throw distribution. The dynamics is 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; set-valued Markov process; ultrafast mixing; combinatorial equilibrium; Gibbs measure