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# Flexagons Lead to a Catalan Number Identity

David Callan
The American Mathematical Monthly
Vol. 119, No. 5 (May 2012), pp. 415-419
DOI: 10.4169/amer.math.monthly.119.05.415
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.05.415
Page Count: 5
Item Type
Article
References
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## Abstract

Abstract Hexaflexagons were popularized by the late Martin Gardner in his first Scientific American column in 1956. Oakley and Wisner showed that they can be represented abstractly by certain recursively defined permutations called pats, and deduced that they are counted by the Catalan numbers. Counting pats by the number of descents yields the identity \documentclass{article} \pagestyle{empty}\begin{document} $$\sum_{k=0}^{n}\frac{1}{2n-2k+1}\binom{2n-2k+1}{k}\binom{2k}{n-k} = C_{n},$$ \end{document} where only the middle third of the summands are nonzero.