Enumerating (Multiplex) Juggling Sequences

Enumerating (multiplex) juggling sequences

We consider the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or 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 ...

Bubblesort and Juggling Sequences

Juggling card sequences

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...

Primitive Juggling Sequences

Over the past two decades, a number of ways have been developed for representing periodic juggling patterns by nite sequences of nonnegative integers. In the most common of these, the entries of the sequence indicate how long the thrown object stays in the air. These juggling sequences have many interesting mathematical properties. In particular, each juggling sequence has associated with it a ...

Enumerating De Bruijn Sequences

A cycle is a sequence taken in a circular order—that is, follows , and are all the same cycle as . Given natural numbers and , a cycle of letters is called a complete cycle [1, 2], or De Bruijn sequence, if subsequences consist of all possible ordered sequences over the alphabet . In 1946, De Bruijn proved [1] (see [2]) that the number of complete cycles, under , is equal to . We propose the ov...