Unlabeled Stamp Foldings

As you’ve seen, a nice problem in combinatorics is to list the number of ways to fold a strip of n stamps into a stack one stamp wide and n stamps tall. For example, if the stamps are different, two stamps can be folded in two ways:

2 stamps, 2 stacks

But if the stamps are blank, there’s really only one shape:

2 stamps, 1 shape

Likewise, three distinct stamps can be folded in six ways:

3 stamps, 6 stacks

But there are only two basic shapes: the first labeled folding is the same as the last one upside down, and the second one is the same as the next three flipped horizontally or vertically or both:

3 stamps, 2 shapes

For four stamps, five basic shapes:

4 stamps, 5 shapes

Five stamps, fourteen shapes:

5 stamps, 14 shapes

There doesn’t seem to be a closed-form formula for computing these counts, either. The first few terms look like the Catalan numbers, but larger numbers don’t match up; see OEIS A001011.

For a 2×2 sheet of stamps, there are 8 distinct foldings, but they’re all the same shape.

2 x 2 blank sheet, only one shape

If you kind of liked this, you’ll kind of love labeled stamp foldings and map foldings.

See Martin Gardner, Wheels, Life and Other Mathematical Amusements, pp. 60–61, 1983.

Figures created with Mathematica 7.

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