Juggling and applications to q-analogues

We consider juggling patterns where the juggler can only catch and throw one ball at a time, and patterns where the juggler can handle many balls at the same time. Using a crossing statistic, we obtain explicit q-enumeration formulas. Our techniques give a natural combinatorial interpretation of the q-Stirling numbers of the second kind and a bijective proof of an identity of Carlitz. By generalizing these techniques, we give a bijective proof of a q-identity involving unitary compositions due to Haglund. Also, juggling patterns enable us to easily compute the Poincare series of the affine Weyl group Ad−1.