Counting mountain-valley assignments for flat folds

Abstract: We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a {\em crease pattern} if it represents the crease lines needed to fold a piece of paper into something. A {\em flat fold} is a crease pattern which lies flat when folded, i.e. can be pressed in a book without crumpling. Given a crease pattern $C=(V,E)$, a {\em mountain-valley (MV) assignment} is a function $f:E\rightarrow {$M,V$}$ which indicates which crease lines are convex and which are concave, respectively. A MV assignment is {\em valid} if it doesn't force the paper to self-intersect when folded. We examine the problem of counting the number of valid MV assignments for a given crease pattern. In particular we develop recursive functions that count the number of valid MV assignments for {\em flat vertex folds}, crease patterns with only one vertex in the interior of the paper. We also provide examples, especially those of Justin, that illustrate the difficulty of the general multivertex case.