Geometric juggling with q-analogues
arXiv:1310.2725 · doi:10.1016/j.disc.2015.02.004
Abstract
We derive a combinatorial equilibrium for bounded juggling patterns with a random, -geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a steady-state distribution containing -rook polynomial coefficients and -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 -analogues. Keywords: juggling pattern; -Stirling number of the second kind; Ferrers board; Markov process; combinatorial equilibrium
14 pages, 3 figures, final version