Journal of Combinatorics

Volume 8 (2017)

Number 3

Guest Editor: Steve Butler (Iowa State University)

Juggling card sequences

Pages: 507 – 539



Steve Butler (Department of Mathematics, Iowa State University, Ames, Ia., U.S.A.)

Fan Chung (Department of Mathematics, University of California at San Diego)

Jay Cummings (Department of Mathematics, California State University, Sacramento, Cal., U.S.A.)

Ron Graham (Department of Computer Science and Engineering, University of California at San Diego)


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 juggling patterns satisfying certain ordering constraints, including where the number of crossings is fixed.


juggling, cards, crossings, enumeration

2010 Mathematics Subject Classification


Steve Butler is partially supported by an NSA Young Investigator Grant.

Received 21 June 2015

Published 21 June 2017