A094504 -id:A094504 - cp's OEIS frontend

A096322 Limiting sequence formed by rows of A094504 read backwards: rightmost floor(n/2)+1 terms of row n in table A094504.

1, 3, 9, 25, 66, 165, 402, 943, 2163, 4835, 10598, 22785, 48215, 100470, 206620, 419662, 842928, 1675487, 3298688, 6436210, 12453352, 23905923, 45550529, 86180937, 161964145, 302447657

Offset: 1

Wouter Meeussen, Jun 27 2004

Same sequence, multiplied by four, occurs in A096272.

a(n) is the number of solid partitions with layer structure an integer partition of (2n-2) in exactly (n-1) parts. - Wouter Meeussen, Mar 12 2025

			For n=3 the a(3)= 9 solid partitions are generated by the integer partitions of (2n-2) in exactly (n-1) parts with parts =1 and duplicate parts deleted, so just {3} and {2} :
 z[{{3}}], z[{{2,1}}], z[{{1,1,1}}], z[{{2},{1}}], z[{{1,1},{1}}], z[{{1},{1},{1}}] and  z[{{2}}], z[{{1,1}}], z[{{1},{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581.

Extended to n=26, Wouter Meeussen, May 23 2025

A096272 Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.

Original entry on oeis.org

1, 0, 4, 0, 6, 4, 0, 10, 12, 4, 0, 13, 30, 12, 4, 0, 18, 70, 36, 12, 4, 0, 19, 142, 94, 36, 12, 4, 0, 24, 274, 234, 100, 36, 12, 4, 0, 19, 501, 534, 258, 100, 36, 12, 4, 0, 18, 872, 1186, 630, 264, 100, 36, 12, 4, 0, 13, 1449, 2486, 1482, 654, 264, 100, 36, 12, 4, 0, 10, 2336, 5080, 3346, 1578, 660, 264, 100, 36, 12, 4

Offset: 1

Wouter Meeussen, Jun 22 2004, Sep 21 2008

Solid partitions of n that fit inside a 4-dimensional k X k X k X k box. Regard solid partitions as safe pilings of boxes in a corner, stacking height does not increase away from the corner and each box contains an integer and this integer too does not increase away from the corner.

If k > 1+(n/2) then T(n,k) = T(n-1,k-1). For large n and k, each row ends as the reverse of 4, 12, 36, 100, 264, 660, 1608, 3772, 8652, 19340, 42392, 91140, 192860, 401880, 836480, ... = 4*A096322(i), i>=1.

			Triangle T(n,k) begins:
  1;
  0,  4;
  0,  6,  4;
  0, 10, 12,  4;
  0, 13, 30, 12,  4;
  0, 18, 70, 36, 12, 4;
  ...
T(16,2) = 1 because only { {{2,2},{2,2}}, {{2,2},{2,2}} } has only two planes, each plane has no more than 2 columns, each column no more than 2 rows and each element is no larger than 2.

Wouter Meeussen, Table of n, a(n) for n = 1..171 (18 rows)
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A007760, A094504, A094508, A096322.

Table[Count[Max[Max@(Flatten@(List@@#)),Max@@Map[Length,#,{-2}],Length/@List@@#,Length[#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]] ,#]&/@Range[n],{n,1,12}]; (* see link for function definition *)

A096573 Number of fixed points of mirroring operation on solid partitions.

Original entry on oeis.org

1, 2, 4, 8, 13, 24, 39, 68, 110, 182, 288, 468, 728, 1150, 1770, 2751, 4175, 6388, 9597, 14495, 21571, 32200, 47498

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link below.

			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.

Wouter Meeussen, Mma functions for plane and solid partitions
Luigi Togliani, Forms of Crossed and Simple Polygons, Science & Philosophy (2019) Vol. 7, No. 2, 71-82.

Cf. A000293, A094504, A094508, A096272, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par],arg_z/;flip[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025  *)

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

A096574 Number of asymmetric solid partitions under mirroring operation.

Original entry on oeis.org

0, 1, 3, 9, 23, 58, 134, 308, 677, 1470, 3106, 6479, 13260, 26827, 53516

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link above.

			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.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096575, A096576, A096577, A096578, A096579, A096580, A096581.

a(n) = (A000293(n) - A096573(n))/2.

A096575 Number of fixed points of solid partitions under rotation operation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 6, 6, 8, 11, 13, 17, 24, 28, 36, 47, 56, 69, 94, 114, 138, 177, 218, 262

Offset: 1

Wouter Meeussen, Jun 27 2004

Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link below.
Is this the same sequence as A002722? - R. J. Mathar, Sep 04 2008 [This still seems to be true even after 20 terms. - N. J. A. Sloane, Feb 05 2025]
Rotation of each of the plane partitions in a solid partition appears to lead to the same count of fixed points as rotating the 3D-partition as a whole. - Wouter Meeussen, Feb 05 2025

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par], arg_z /;turn[arg]==arg],{n,20}, {par, IntegerPartitions[n]}]

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

a(24)-a(25) from Wouter Meeussen, Jul 27 2025

A096577 Number of fixed points of solid partitions under 'time-lapse' operation.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 1, 5, 0, 7, 1, 7, 0, 14

Offset: 1

Wouter Meeussen, Jun 27 2004

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three planes to project into, together with the starting solid partition, that makes four 'isomers'.

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}].

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096578, A096579, A096580, A096581, A119266.

(* See link above. *)
Tr/@Table[Count[solidformBTK[par],arg_z/;lapse[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025 *)

a(16)-a(23) from Wouter Meeussen, Mar 19 2025

A096576 Number of solid partitions asymmetric under rotation operation.

Original entry on oeis.org

0, 1, 3, 8, 19, 46, 101, 226, 486, 1038, 2163, 4471, 9077, 18260, 36258

Offset: 1

Wouter Meeussen, Jun 27 2004

Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link above.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096577, A096578, A096579, A096580, A096581.

Mathematica
```
(* See A096575. *)
```

a(n) = (A000293(n) - A096575(n))/3.

A096578 Number of solid partitions with period (cycle length) two under 'time-lapse' operation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 6, 7, 11, 15, 25, 33, 48, 65

Offset: 1

Wouter Meeussen, Jun 27 2004

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three plane to project into, together with the starting solid partition, that makezs four 'isomers'.

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096577, A096579, A096580, A096581.

Mathematica
```
(* See link above. *)
```

A096581 Number of solid partitions non-symmetric under L^2 (L= 'time-lapse' symmetry operation) on solid partitions.

Original entry on oeis.org

0, 2, 4, 12, 28, 68, 150, 336, 724, 1550, 3234, 6688, 13590, 27354, 54334

Offset: 1

Wouter Meeussen, Jun 27 2004

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096577, A096578, A096579, A096580.

Mathematica
```
(* See link above. *)
```

By definition, A000293(n) = A096580(n) + 2*a(n).

A089924 Number of plane partitions of n with 3 or more columns.

Original entry on oeis.org

0, 0, 1, 3, 8, 19, 41, 85, 167, 319, 588, 1066, 1880, 3272, 5588, 9431, 15686, 25841, 42070, 67926, 108634, 172467, 271643, 425109, 660756, 1021170, 1568905, 2398059, 3646437, 5519025, 8314623, 12473538, 18634748, 27731820, 41113453, 60735830, 89411503, 131193675

Offset: 1

Wouter Meeussen, Jan 11 2004

nonn

Because of symmetry of the 3-dimensional-Ferrers plots, this sequence also counts the plane partitions with 3 or more rows and the plane partitions with maximal element >= 3

			a(4)=3:
  1111 111 211
  .... 1.. ...

Alois P. Heinz, Table of n, a(n) for n = 1..5000
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000219, A000991, A091357.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
      *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, infinity)-b(n,2):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018

(* planepartitions[] : see link *); Table[Count[planepartitions[n], q_ /; Length[First[q]] >= 3], {n, 12}]

a(21)-a(27) from Vaclav Kotesovec, May 05 2018

a(28)-a(38) from Alois P. Heinz, Sep 24 2018

cp's OEIS Frontend

A096322 Limiting sequence formed by rows of A094504 read backwards: rightmost floor(n/2)+1 terms of row n in table A094504.

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A096272 Triangle read by rows: T(n,k) counts solid partitions of n such that the maximum of planes, rows, columns and values is k.

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A096573 Number of fixed points of mirroring operation on solid partitions.

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A096574 Number of asymmetric solid partitions under mirroring operation.

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A096575 Number of fixed points of solid partitions under rotation operation.

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A096577 Number of fixed points of solid partitions under 'time-lapse' operation.

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A096576 Number of solid partitions asymmetric under rotation operation.

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A096578 Number of solid partitions with period (cycle length) two under 'time-lapse' operation.

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