## Geometric juggling with *q*-analogues

*Discrete Mathematics*, 338(7):1067–1074, 2015.- doi:10.1016/j.disc.2015.02.004
- arXiv:1310.2725.

### Abstract

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