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<DIV>I'm wondering if anybody has a table of polyominos classified by their orbits under permutations. If not, it should be easy to modify a polyomino generating program to generate such a table. For example, the entries for pentominos might look like:</DIV>
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<DIV>5 [1^5]</DIV>
<DIV>2 [1^5] [1^3,2]</DIV>
<DIV>3 [1^5] [1,2^2]</DIV>
<DIV>1 [1^5]^2 [1,2^2]^2</DIV>
<DIV>1 [1^5] [1^3,2]^2 [1,2^2]^3 [1,4]^2</DIV>
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<DIV>The first line here represents the five asymmetric pentominos, which have all five squares fixed under their only symmetry, the identity. The last line is the "X" pentomino: the identity fixes all five squares; the horizontal and vertical flips fix one axis and swap one pair; the diagonal flips and the 180 degree rotation fix the central square and pair up the others; and the two 90 degree rotations fix the center and put the other four into a single cycle.</DIV>
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<DIV>My primary interest in this is for the "piled" polycubes I introduced recently, but they are also useful for polyomino counting problems. For example, the number of two-color polyominos starts:</DIV>
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<DIV>1,2,3,12,54,300,1864,12492</DIV>
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<DIV>which is not in the OEIS. (This is based on a manually generated table up to n=7. One can also use the information in this table to generate the number of fixed polyominos of size n, which checks, so I am fairly confident that this is correct.)</DIV>
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<DIV>Needless to say, I would like this table extended as far as I can get it.</DIV>
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<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV></DIV></DIV>
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