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A Combinatorial Problem:
How many different n-level mazes are there?

Among the classical and medieval maze-types there are examples of different simple alternating transit mazes with the same number of levels. For instance
these three 12-level maze-paths all occur. This raises the question: How many different mazes are there with n levels?

Now it is possible to list all of the possible s.a.t. mazes of a given depth n, by looking at all the permutations of 0,...,n and discarding those which do not meet the three conditions which guarantee that a sequence corresponds to a maze. In fact, since odds and evens cannot mix in the permutation, and since 0 and n must stay fixed, it is enough to look at all pairs of permutations, one of the odd integers 1,...,n-1 (assuming n even), and one of the even integers 2,...,n-2. For example, to construct all possible 8-level mazes, one must consider pairs of permutations of 1,3,5,7 and of 2,4,6. There are 24 x 6 = 144 such pairs, so this can easily be done by hand. Here is what one finds for depths 1 through 8. In this list, the non-"interesting" mazes (those can be constructed from shallower mazes by adding trivial levels at the top and/or bottom) have been omitted. This figure shows the nuclei and rectangular mazepaths of these mazes; only one representative of each dual pair is drawn. See From rectangular path to nucleus for how these forms are related (added 3/14/2024).

Corrections (03/02/2024, 03/09/2024, 03/14/2024). In "The interesting mazes of 4,6,7 and 8 levels," 034562718 should be 034567218, and the dual of 056743218 is 076541238; also the nucleus for 056723418 is wrong: its right side should be the same as the right side of 072345618 or 034562718. Thanks to Dani Novak.

Here are the data in tabular form:

Depth Number of mazes Interesting mazes
 1 1 none
 2 1 none
 3 1 none
 4 2 03214
 5 3 none
 6 8 0523416 0543216
 0345216 0541236
 7 14 03216547 
 8 42 032147658 034567218 034765218 
 036547218 054367218 056723418 
 056741238 056743218 072345618 
 072365418 072543618 074325618 
 074561238 074563218 076123458 
 076125438 076143258 076321458 
 076345218 076523418 076541238 
 076543218 

For depth 9 one would have to consider 24 x 24 = 576 permutations, and the possibilities of error increase. The following numbers were calculated using computers.

 n M(n) I(n)
 1 1 0 
 2 1 0 
 3 1 0 
 4 2 1
 5 3 0 
 6 8 4 
 7 14 1 
 8 42 22 
 9 81 11 
 10 262 142 
 11 538 95 
 12 1828 1014 
 13 3926 808 
 14 13820 7796
 15 30694 6980 
 16 110954 63386 
 17 252939 61725 
 18 933458 538534 
 19 2172830 558853 
 20 8152860 4740658 
 21 19304190 5171300 
 22 73424650 42969130
 24 678390116
 26 6405031050 
 28 61606881612 
 30 602188541928
 32 5969806669034

Here M(n) and I(n) are the numbers of mazes and of interesting mazes of depth n. The values for 24, 26 and 28 were obtained by Jim Reeds of Bell Labs by a method which does not seem to apply to odd depth-numbers. The values for 30 and 32 are reported in Lando and Zvonkin's 1993 paper as due to V. R. Pratt.

The interesting 9-level mazes are shown this figure by their rectangular maze-paths; again, only one representative of each dual pair is drawn.

The next figure shows the maze paths of the 5 self-dual very-interesting mazes of depth 10; these are mazes which also have no internal sequences of trivial levels.

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How are these numbers calculated?

Why did I get interested in this problem?

Where else do these numbers occur?

What else is known about the maze numbers?

References

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