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Number of fixed points of mirroring operation on solid partitions.
9

%I #8 Apr 23 2020 20:44:43

%S 1,2,4,8,13,24,39,68,110,182,288,468,728,1150,1770

%N Number of fixed points of mirroring operation on solid partitions.

%C Uses function "solidformBTK" from link above.

%H Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/SolidPartitions.txt">Solid Partitions Mathematica functions</a>

%H Luigi Togliani, <a href="https://doi.org/10.23756/sp.v7i2.473">Forms of Crossed and Simple Polygons</a>, Science & Philosophy (2019) Vol. 7, No. 2, 71-82.

%e Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] mirrors into [{{3, 3}, {1}, {1}, {1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by mirroring each layer as a plane partition.

%t flip[pili_List] := Module[{wide, it}, wide =Length[pili[[1]]]; it = Join[ #, Table[0, {wide - Length[ # ]}]] & /@ pili; DeleteCases[ Transpose[it], 0 | {}, -1]]; Table[sn = Sort@Flatten[solidformBTK /@ Partitions[n]]; Frequencies[Length /@ ToCycles[Ordering[Map[flip@ # &, sn, {2}]]] ], {n, 1, 15}]

%Y Cf. A000293, A094504, A094508, A096272, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581.

%K nonn,more

%O 1,2

%A _Wouter Meeussen_, Jun 27 2004