The stamp folding problem asks for the number of ways to fold a strip of 
stamps, where 
is a positive integer,
into a flat stack without self-intersection, with variants determined by which features
of a folding are distinguished. For stamps labeled consecutively from 1 to 
, let 
count the distinct top-to-bottom label orders, with the
same labels on both sides and left-right reflections
identified. Let 
count the unlabeled folding shapes, identifying reflections
and reversal of the labels, and let 
count the symmetric labeled
foldings whose label orders are unchanged by reversing the stack and replacing each
label 
by 
(Koehler 1968, pp. 136-138; Legendre 2014, pp. 276-279). The following
table summarizes these values.
 |  |  |  |
| Sloane | A001010 | A001011 | A000136 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 1 | 2 |
| 3 | 2 | 2 | 6 |
| 4 | 4 | 5 | 16 |
| 5 | 6 | 14 | 50 |
| 6 | 8 | 38 | 144 |
| 7 | 18 | 120 | 462 |
| 8 | 20 | 353 | 1392 |
| 9 | 56 | 1148 | 4536 |
| 10 | 48 | 3527 | 14060 |
For labeled stamp foldings, Sakai (2026) described a boundary-based decomposition in which the second stamp is fixed as a boundary. Under suitable conditions, the portions above and below this boundary can be treated independently, giving a product structure that explains the appearance of recurrence relations. The decomposition is structural rather than a closed counting formula, and it does not extend directly to variants without a fixed boundary or additional symmetries.
