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<DIV>I've just gotten interested in what I'm calling "piled" polyominos. A piled polyomino can be thought of as a polyomino that is subject to "gravity" - it rests on a baseline, and no square in the polyomino can be on empty space.</DIV>
<DIV> </DIV>
<DIV>A 2D piled polyomino can be represented by a finite sequence of positive integers, representing how many squares are in the pile. The total of the sequence is then the size of the polyomino. It follows that the number of "fixed" 2D polyominos of size n is 2^(n-1).</DIV>
<DIV> </DIV>
<DIV>The next case is when reversals are allowed. In this case, the number of size n is A005418(n) = 2^(n-2) + 2^([n/2]-1).</DIV>
<DIV> </DIV>
<DIV>It starts to get interesting when we allow rotations. Many piled polyominos cannot be rotated, but some can. Allowing rotations but not reflections, we get a sequence that starts:</DIV>
<DIV> </DIV>
<DIV>1,1,3,5,11,24,51,110</DIV>
<DIV> </DIV>
<DIV>(if I haven't made any mistakes), and allowing both rotations and reflections, we get:</DIV>
<DIV> </DIV>
<DIV>1,1,2,4,7,15,29,62</DIV>
<DIV> </DIV>
<DIV>Neither of these is in the OEIS; I will submit them, but I'm hoping someone can compute a few more terms first.</DIV>
<DIV> </DIV>
<DIV>Turning to the 3D case, I have submitted the following sequence (with an A-number from the dispenser):</DIV>
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<DIV>%I A113174<BR>%S A113174 1 3 11 44 184 792 3484 15592 70745 324561 1502511 7007929 32892778 155221536<BR>735915652 3503270920 16737092 80218277681 385574074383 1858059853316<BR>8974761939239 43441619693731 210682920968681 1023583453675349 4981114010928174<BR>2426349891204 3132240320982559396171278383569<BR>%N A113174 Fixed 3D Piled Polyominos - polycubes with fixed orientation, with no cubes "sitting on air".<BR>%F A113174 a(n) = sum_{m=1}^n A001168(m)*C(n-1,m-1)<BR>If both sequences shifted left, binomial transform of A001168.<BR>%e A113174 For n = 4, there are 4 orientations of the angled tricube excluded: those which set it on a point; this leaves 8 orientations of the angled tricube and 3 of the straight tricube.<BR>%Y A113174 cf A001168<BR>%O A113174 1<BR>%K A113174 ,nonn,<BR>%A A113174 Frank Adams-Watters (FrankTAW@Netscape.net), Oct 15 2005<BR></DIV>
<DIV>I wanted to add a reference to the sequence for fixed polycubes, but it doesn't appear to be in the database! It starts:</DIV>
<DIV> </DIV>
<DIV>1,3,15,74</DIV>
<DIV> </DIV>
<DIV>Surely somebody can extend this one.</DIV>
<DIV> </DIV>
<DIV>There are 4 sequences of piled 3D polys that need extending:</DIV>
<DIV> </DIV>
<DIV>allowing only planar rotations:</DIV>
<DIV> </DIV>
<DIV>1,2,4,15</DIV>
<DIV> </DIV>
<DIV>allowing arbitrary planar transformations:</DIV>
<DIV> </DIV>
<DIV>1,2,3,12</DIV>
<DIV> </DIV>
<DIV>allowing spatial rotations:</DIV>
<DIV> </DIV>
<DIV>1,1,2,8</DIV>
<DIV> </DIV>
<DIV>and allowing arbitrary isometric transformations:</DIV>
<DIV> </DIV>
<DIV>1,1,2,7</DIV>
<DIV> </DIV>
<DIV>These clearly need more terms.</DIV>
<DIV> </DIV>
<DIV>One other point: the index entries for "Polycube" and "Polyomino - 3D" reference only A000162, which is polycubes invariant only under spatial rotations. In my opinion, the true primary polycube sequence is A038119, which counts mirror images as the same. At minimum, it should be included in these index entries.</DIV>
<DIV> </DIV>
<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV></DIV></DIV></DIV><!-- end of AOLMsgPart_2_3dee0389-bee6-4178-88d3-8a7e8dcc7fd7 --></DIV></DIV>
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