rotated & reflected safe pilings (=plane partitions)

[vdmcc]

Imagine a room, where boxes (cubes) are to be safely piled in the far left
corner.
For stability reasons, stacking height should not increase away from the
corner.in how many ways can n boxes be piled?

Table[Plus@@(Length[pilings[#]]&/@Partitions[w]),{w,16}]
{1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297}
=A000219
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Q: in how many ways, taking x-y reflections as equivalent?
Table[Length at Union[ Flatten[ {#,flip[#]}&/@
(Flatten[pilings/@Partitions[n],1]),1] ] ,{n,16}]
{1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297}
=A000219
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Q: in how many ways, taking x-y-z-x rotations as equivalent?
Table[Length@ Union[ rotapiling/@
(Flatten[pilings/@Partitions[n],1])  ],{n,16}]
{1,1,2,5,8,16,30,54,94,168,287,493,831,1391,2293,3769}
I am sorry, but the terms
2,5,8,16,30,54
do not match anything in the table
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Q: in how many ways, taking both symmetries as equivalent?

Table[Length@  Union[  majorpiling/@
(Flatten[pilings/@Partitions[n],1])  ]   ,{n,16}]

{1,1,2,4,6,11,19,33,55,95,158,267,442,731,1193,1947}
=A000786
ID Number: A000786 (Formerly M1020 and N0383)
Sequence:  1,1,2,4,6,11,19,33,55,95,158,267,442,731,1193,1947
Name:      Planar partitions of n.
References P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press,
London and New York, Vol.
           1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
Keywords:  nonn
Offset:    1
Author(s): njas
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