Enumerating (multiplex) juggling sequences

TL;DR

This paper develops a recursive method for enumerating periodic multiplex juggling sequences by translating the problem into a matrix selection task with specific sum constraints, generalizing previous work.

Contribution

It introduces a new recursive approach for counting multiplex juggling sequences, extending prior results to more complex initial states.

Findings

Derived a recursion for counting sequences

Established equivalence to matrix sum problems

Generalized earlier enumeration methods

Abstract

We consider the problem of enumerating periodic $\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\sigma$ is the initial state (or {\em landing schedule}) of the balls. We first show that this problem is equivalent to choosing 1's in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Fan Chung and Ron Graham.