Juggling card sequences

Abstract

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.

Counting Prime Juggling Patterns

“…. , C b } placed consecutively in some order with the card C b used at least once (see [2,6]). This can be seen by noting that placing the cards together gives a variant of the juggling diagram (i.e., the diagram that traces out the path of the balls over a time window of period n) from which the juggling pattern can be recovered.…”

“…Juggling has many interesting connections with combinatorics (see [1,2,6]). There are several ways to describe juggling patterns, and each description gives some information about various properties of the patterns.…”