Generalized Eulerian Numbers and Multiplex Juggling Sequences
Esther Banaian · 2016
We consider generalizations of both juggling sequences and non-attacking rook placements. We demonstrate the important connection between these objects, and also propose a generalization of the Eulerian numbers. These generalizations give rise to several interesting counting problems, which we explore. 1 History of Mathematics of Juggling Juggling as an activity has been around for the last 4000 years. There are many ancient records, both written documents and depictions, of people performing amazing juggling feats (for more on this, see [19]). Siteswap, the first mathematical way to describe juggling patterns, was formalized around 1985. Several papers appeared in the earlier years which investigated some of the possible ways to express juggling sequences, but they lacked consistency. In [20], published in 1982, Walker uses the idea of throw heights to invent new juggling patterns, even though the formal concept of a “throw height” had not yet been fully explored in the juggling community. Two years later, Buhler and Graham explored the physical dynamics of several commonly known juggling patterns [4]. Their paper considers many parameters of juggling. Several of these parameters, such as dwell time of a ball in one’s hand, do not reappear in any of these authors’ future juggling papers, and we assume they found the analysis of all these different factors too complicated to be useful. In 1985, though, the juggling community became more cohesive, as three groups separately developed the beginnings of the modern mathematics of juggling [19]. These juggling founders included Paul Klimek in Santa Cruz, California, Bruce Tiemann and Bengt Magnusson at Caltech, and Colin Wright at Cambridge University. A record of some early emails from Magnusson about his siteswap system, as well as his computer program to generate siteswaps, can be found at [14]. Wright provides a thorough introduction to the concept of siteswap in both [11] and [21]. Most of this