Juggling card sequences

TL;DR

This paper explores the algebraic and combinatorial properties of juggling patterns represented by card sequences, focusing on enumeration under specific ordering constraints and fixed crossings.

Contribution

It introduces new methods for enumerating juggling patterns with ordering constraints and fixed crossings, linking juggling sequences to various combinatorial structures.

Findings

Enumeration formulas for constrained juggling patterns

Connections to Stirling numbers, Dyck paths, and Narayana numbers

Analysis of patterns with fixed crossings

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.