2,238 research outputs found
Descent polynomials for permutations with bounded drop size
Motivated by juggling sequences and bubble sort, we examine permutations on
the set {1,2,...,n} with d descents and maximum drop size k. We give explicit
formulas for enumerating such permutations for given integers k and d. We also
derive the related generating functions and prove unimodality and symmetry of
the coefficients.Comment: 15 page
Toss and Spin Juggling State Graphs
We review the state approach to toss juggling and extend the approach to spin
juggling, a new concept. We give connections to current research on random
juggling and describe a professional-level juggling performance that further
demonstrates the state graphs and their research.Comment: 8 pages, 10 figures, to appear in the Proceedings of Bridges 201
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 Chung and Graham
A Zeta Function for Juggling Sequences
We give a new generalization of the Riemann zeta function to the set of b-ball juggling sequences. We develop several properties of this zeta function, noting among other things that it is rational in b−s. We provide a meromorphic continuation of the juggling zeta function to the entire complex plane (except for a countable set of singularities) and completely enumerate its zeroes. For most values of b, we are able to show that the zeroes of the b-ball zeta function are located within a critical strip, which is closely analogous to that of the Riemann zeta function
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