question about parallel sequence making prime
Jeffrey Shallit
elvis at graceland.math.uwaterloo.ca
Wed Feb 21 15:54:13 CET 2007
sequences cannot be permutations of each other.
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Date: Wed, 21 Feb 2007 09:30:41 -0800
From: "Jonathan Post" <jvospost3 at gmail.com>
To: seqfan <seqfan at ext.jussieu.fr>, "Forrest Bishop" <forrestb at ix.netcom.com>,
jvospost2 at yahoo.com
Subject: Lattice Animals with self-avoiding perimeters
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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 permieter 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, 21, ...
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, 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 quare tetromino with two monominoes sticking out but
not coinciding to a 2x3).
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 it's 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
uniquen 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 flippwd 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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