Lattice Animals with self-avoiding perimeters
Jonathan Post
jvospost3 at gmail.com
Thu Feb 22 05:18:20 CET 2007
*** corrected, extended ***
Following up on Frank TAW's comment about holes in A057730, we may define:
a(n) = the number of nonisomorphic lattice animals (may be more than
one disjoint polyomino) whose perimeter is a self-avoiding loop (holes
allowed), ordered by perimeter = 2n. (flipping over is the same
polyomino).
I count this as: 0, 1, 1, 4, 7, 22, 38, ...
P=4 is the monomino.
P=6 is the domino.
P=8 is: L-triomino, straight triomino, square tetromino, and two
disjoint monominoes.
P=10 is straight tetromino, L-tetromino, T-tetromino, Z-tetromino,
domino+monomino, P-pentomino, 2x3 hexomino.
P=12 is 11 of the pentominoes (excluding the P), square tetromino +
monomino, straight triomino + straight triomino, L- triomino +
straight triomino, 3 disjoint monominoes, and 7 of the hexominoes (the
ones which are a square tetromino with two monominoes sticking out but
not coinciding to a 2x3).
P=14 is the other 27 hexominoes,
10+4 partitions: the 7 with P=10 each with an additional disjoint monomino,
8+6 partitions: the 4 with P=8 each with an additional disjoint domino,
total 27+7+4 = 38.
Frank TAW's hole question is simplified here, in that we exclude the
smallest holey polyomino, as that heptomino does not have a
self-avoiding perimeter. But we do include the holey octomino with a
monomino square. Note that its perimeter is a 3x3 square with a
smaller 1x1 inside, which is the same length but distinct from 3x3 +
1x1 (the square nonomino with disjoint monomino).
There is a recurrence here based on partitioning the perimeter into
even terms >2, i.e. 14 = 14 = 10+4 = 8+6. This is initialized by the
unique solutions for P=4 and P=6. The matter of the two P=16 with
hole versus with the hole turned into an external monomino shows that
there are topological constraints on the partition.
Similar sequences for poly-polyomones which are not flipped over, and
another for fixed lattice animals.
For convex polyominoes there is lovely known closed-form count by
perimeter. But this is different.
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