Flexagons yield a curious Catalan number identity

TL;DR

This paper explores the combinatorial properties of hexaflexagons, demonstrating a new Catalan number identity through recursive permutation representations called pats.

Contribution

It introduces a novel identity involving Catalan numbers derived from counting specific permutation structures related to hexaflexagons.

Findings

Derived a new Catalan number identity involving summation over pats

Connected hexaflexagon combinatorics to permutation descent counts

Identified that only the middle third of summands are nonzero in the identity

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 number of descents yields the curious identity Sum[1/(2n-2k+1)binom{2n-2k+1}{k}binom{2k}{n-k},k=0..n] = C(n), where only the middle third of the summands are nonzero.