The Stamp Folding Problem From a Mountain-Valley Perspective: Enumerations and Bounds

A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We consider this problem with the additional restriction of fixing the mountain-valley assignment of each crease in the stamp pattern. We provide a closed form for counting the number of legal foldings on specific patterns of mountain-valley assignments, including a surprising appearance of the Catalan numbers. We construct upper and lower bounds for the number of ways to fold a given mountain-valley assignment on the strip of stamps. Lastly, we provide experimental evidence towards more general results.