A flexagon is an object created by folding a piece of paper along certain lines to form loops. By manipulating the folds, it is possible to hide and reveal different faces.
Oakley and Wisner (1957, pp. 146-147 and 151-153) describe abstract hexaflexagon configurations using recursively defined permutations
called pats. A one-element permutation is a pat.
A permutation of two or more elements is a pat if
it has a unique split into two nonempty blocks, every element of the first block
exceeds every element of the second, and the reverse of each block is a pat after
its entries are relabeled in increasing order. Its degree is its number of elements.
An abstract configuration is an ordered pair of pats,
and its order 
is the sum of their degrees. There are 
such configurations, where 
is a Catalan number, for
. This counts configurations before
identifying those related by pinches or rotations; it does not count the physical
faces or reachable states of one fixed paper model.
