[seqfan] Tiling with Necklaces
Ron Hardin
rhhardin at att.net
Sat Jul 18 18:29:55 CEST 2015
Consider 3X3 binary arrays with constrained perimeters (so center element free), constrained to be one of several necklaces.
How many T(n,k) nXk binary arrays can you tile with this or that set of necklaces?
Up to 3-necklace patterns, there aren't that many that don't go periodic along rows and columns of T(n,k), i.e. that keep growing as n or k increase.
So far only these (approximately determined by heuristic) necklace perimeter sets have yet to be done
00000000 00000001 00001011
00000000 00000001 00010001
00000001 00000101 00001001
00000001 00000011 00001001
00000000 00000001 00010101
00000001 00000011 00001111
00000000 00000001 00001111
00000001 00000011 00001011
00000101 00010001 00010101
00000001 00000011 00010001
00000001 00000011 00010011
00000001 00000011 00010101
00010101 00101011 01010101
00000001 00000011 00100101
00010101 00100101 01010101
00000001 00001001 00100101
00010001 00010101 01010101
00000001 00010101 01010101
00001011 00010101 01010101
00000011 00010101 01010101
00000001 00000101 00001011
00000001 00000101 00000111
Heuristic - try every set of 3 modulo rotation, compute T(4,4), sort the list and do them from largest to smallest, one set for each value of T(4,4) (or T(4,4)+1).
Stop doing cases when T(n,k) for some set goes periodic, which the last line does.
(Necklaces are not automatically flipped, though almost all are symmetric under flip anyway) rhhardin at mindspring.com
rhhardin at att.net (either)
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