  Juggling patterns can be described by a sequence of cards which keep track of
the relative order of the balls at each step. This interpretation has many
algebraic and combinatorial properties, with connections to Stirling numbers,
Dyck paths, Narayana numbers, boson normal ordering, arc-labeled digraphs, and
more. Some of these connections are investigated with a particular focus on
enumerating juggling patterns satisfying certain ordering constraints,
including where the number of crossings is fixed.
