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2013b.00743 Callan, David Flexagons lead to a Catalan number identity. Am. Math. Mon. 119, No. 5, 415-419 (2012). 2012 The Mathematical Association of America, Washington, DC EN K25 Hexaflexagons pats

doi:10.4169/amer.math.monthly.119.05.415

Summary: Hexaflexagons were popularized by the late {\it M. Gardner} in his first column in Scientific American [``Hexaflexagons and other mathematical diversions" (1956)]. {\it C. O. Oakley} and {\it R. J. Wisner} [Am. Math. Mon. 64, 143--154 (1957; Zbl 0077.01901)] 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 $$\sum^n_{k=0}\frac {1}{2n - 2k + 1}\binom {2n - 2k + 1}{k}\binom {2k}{n - k} = C_{n},$$ where only the middle third of the summands are nonzero.