10 research outputs found
Juggling and vector compositions
AbstractA considerable amount of interest has arisen pertaining to the mathematics of juggling. In this paper, we use techniques similar to those of Ehrenborg and Readdy (Discrete Math. 157 (1996) 107) to enumerate two special classes of juggling patterns. Both classes are enumerated by a product of q-binomial coefficients; the second class is a generalization of the first. Using the first class of patterns, we give a bijective proof of an identity of Haglund (Compositions, rook placements, and permutations of vectors, Doctoral Dissertation, University of Georgia, Athens, GA, 1993) involving vector compositions. We define a generalized vector composition and provide a bijective proof of an identity dual to Haglund's using these generalized vector compositions
Multivariate Juggling Probabilities
We consider refined versions of Markov chains related to juggling introduced
by Warrington. We further generalize the construction to juggling with
arbitrary heights as well as infinitely many balls, which are expressed more
succinctly in terms of Markov chains on integer partitions. In all cases, we
give explicit product formulas for the stationary probabilities. The
normalization factor in one case can be explicitly written as a homogeneous
symmetric polynomial. We also refine and generalize enriched Markov chains on
set partitions. Lastly, we prove that in one case, the stationary distribution
is attained in bounded time.Comment: 28 pages, 5 figures, final versio
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
Enumeration of multiplex juggling card sequences using generalized q-derivatives
In 2019, Butler, Choi, Kim, and Seo introduced a new type of juggling card
that represents multiplex juggling patterns in a natural bijective way. They
conjectured a formula for the generating function for the number of multiplex
juggling cards with capacity 2. In this paper we prove their conjecture. More
generally, we find an explicit formula for the generating function with any
capacity. We also find an expression for the generating function for multiplex
juggling card sequences by introducing a generalization of the q-derivative
operator. As a consequence, we show that this generating function is a rational
function.Comment: 17 pages, 4 figure
COMBINATORIAL ASPECTS OF EXCEDANCES AND THE FROBENIUS COMPLEX
In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter