Stamp folding patterns and a boundary-based counting approach

The stamp folding problem asks for the number of distinct ways to fold a strip of stamps without self-intersection. In its standard and most widely studied formulation, the problem is considered in the labeled setting. In this paper, we study the labeled stamp folding problem from a structural point of view and describe a boundary-based decomposition centered at the second stamp. By fixing the second stamp as a boundary, a folding configuration can be separated into upper and lower parts that, under suitable conditions, can be treated independently. This viewpoint leads to a natural product structure and provides an explanation for the appearance of recurrence-type relations in this setting. The main contribution of this work is not the introduction of new counting techniques, but a clarification of the combinatorial mechanism underlying such recurrences. The formulation also highlights inherent limitations of the approach, explaining why analogous decompositions do not directly extend to variants of the problem without a fixed boundary or with additional symmetries.