  In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card
that represents multiplex juggling patterns in a natural bijective way. They
conjectured a formula for the generating function for the number of multiplex
juggling cards with capacity 2. In this paper we prove their conjecture. More
generally, we find an explicit formula for the generating function with any
capacity. We also find an expression for the generating function for multiplex
juggling card sequences by introducing a generalization of the q-derivative
operator. As a consequence, we show that this generating function is a rational
function.
