Enumerating (multiplex) 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...

Excedances of affine permutations

We introduce an excedance statistic for the group of affine permutations S̃n and determine the generating function of its distribution. The proof involves working with enumerating lattice points in a skew version of the root polytope of type A. We also show that the left coset representatives of the quotient S̃n/Sn correspond to increasing juggling sequences and determine their Poincaré series.

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. Résumé. Motivés par les “suites de jonglerie” et le tri à bull...

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 ...

Bubblesort and Juggling Sequences