Necklaces, convolutions, and X + Y
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
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In the late 1980s it was shown that juggling patterns can be described by strings of numbers with fascinating combinatorial properties that have since then been studied by many mathematicians and computer scientists. In this paper we propose to study juggling patterns from a pattern matching point of view. Inspired by pattern matching algorithms based on convolution, we propose a fast algorithm for finding transitions between juggling states. Apart from being a fun application of pattern matching theory, it provides a practical tool in the experimental design of (large) juggling patterns. We also provide a compact formula for counting the number of transitions of a given length between two juggling states.