The Stamp Folding Problem From a Mountain-Valley Perspective

TL;DR

This work studies the fixed mountain-valley foldings of a labeled $1 \times n$ stamp strip by counting $c(\mu)$, the number of valid foldings consistent with a given MV assignment $\mu$. Building on a meander-based, layer-ordering framework, the authors derive a closed form for two families of MV patterns: (i) two-block assignments $M^aV^b$, which exhibit polynomial growth in $n$ and, for $a>b+1$ and $b>4$, yield $c(M^aV^b)=ab+b^2-b+1$ with several edge-case corrections; (ii) 2-alternating assignments $(M^2V^2)^m$, where $c=(C_{loor{m/2}+1})(C_{loor{m/2}+1})$ ties the count to Catalan numbers and a lattice-walk interpretation, implying exponential growth. The paper connects these results to Catalan structures, provides bounds and experimental insights for general MV assignments, and outlines conjectures on asymptotics and maximal-folding patterns, highlighting open questions and directions for extending the analysis to higher grids such as $2\times n$. The work advances understanding of fixed MV foldings and offers concrete tools (meander view, walk-based encodings) for analyzing specific MV patterns.

Abstract

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 describe results on upper and lower bounds for the number of ways to fold a given mountain-valley assignment on the strip of stamps, provide experimental evidence suggesting more general results, and include conjectures and open problems.

Figures (7)

• Figure 1: $c(M^2 V^2) = 4$
• Figure 3: The various cases of Theorem \ref{['thm:MaVb']}.
• Figure 4: An example of a 2-alternating assignment folding of $(M^2V^2)^4 M^2$.
• Figure 5: Illustration of the proof of Lemma \ref{['lem:m-first-bot']}. Not all arcs are shown.
• Figure 6: With only uncovered bottom arcs shown, at $i$ there are $4$ potential gaps to the left and $3$ to the right. If $(i, i+1)$ is a mountain arc then $f(i) = 4$ and $b(i) = 3$. Otherwise $f(i) = 3$ and $b(i) = 4$.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Remark 2.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Remark 3.1
• Theorem 3.2