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