A119266 -id:A119266 - cp's OEIS frontend

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

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

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

A119269 Table by antidiagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 6, 6, 4, 2, 1, 1, 1, 8, 11, 7, 4, 2, 1, 1, 1, 12, 19, 13, 7, 4, 2, 1, 1, 1, 16, 33, 25, 14, 7, 4, 2, 1, 1, 1, 22, 55, 49, 27, 14, 7, 4, 2, 1, 1, 1, 29, 95, 93, 55, 28, 14, 7, 4, 2, 1, 1, 1, 40, 158, 181, 111, 57, 28, 14, 7, 4, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, May 11 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Transposed table is A119338. - Max Alekseyev, May 14 2006

			Table starts:
  1, 1,  1,  1,  1
  1, 1,  1,  1,  1
  1, 2,  2,  2,  2
  1, 3,  4,  4,  4
  1, 4,  6,  7,  7
  1, 6, 11, 13, 14

Columns A005987, A000786, A119266, A119267; diagonal A119268. Cf. A096751, A119270.

Cf. A119339, A119340, A119341, A119342.

a(n,m) = a(n,n-2) for m >= n-1.

More terms from Max Alekseyev, May 14 2006

A119338 Table by antidiagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 4, 1, 1, 1, 2, 4, 6, 6, 1, 1, 1, 2, 4, 7, 11, 8, 1, 1, 1, 2, 4, 7, 13, 19, 12, 1, 1, 1, 2, 4, 7, 14, 25, 33, 16, 1, 1, 1, 2, 4, 7, 14, 27, 49, 55, 22, 1, 1, 1, 2, 4, 7, 14, 28, 55, 93, 95, 29, 1, 1, 1, 2, 4, 7, 14, 28, 57, 111, 181, 158, 40, 1

Offset: 1

Max Alekseyev, May 15 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

			Table starts:
  1, 1, 1, 1, 1,  1, ...
  1, 1, 2, 3, 4,  6, ...
  1, 1, 2, 4, 6, 11, ...
  1, 1, 2, 4, 7, 13, ...
  1, 1, 2, 4, 7, 14, ...
  ...

Rows: A000012, A046682, A000786, A119266, A119267, A119340, A119341, A119342 stabilize to A119268. Transposed table is A119269. Cf. A119339, A119270, A118364, A118365.

A119267 Number of 4-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 27, 55, 111, 232, 486, 1039, 2226, 4820, 10449, 22727, 49354, 107117, 231774, 500040, 1074476, 2299589, 4899650

Offset: 0

Franklin T. Adams-Watters, May 11 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Cf. A119269, A000334.

Cf. A119266, A119338, A119340, A119341, A119342.

a(9)-a(20) from Max Alekseyev, May 16 2006

a(21)-a(23) from Max Alekseyev, Mar 20 2025

A119340 Number of 5-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 57, 117, 251, 543, 1209, 2724, 6251, 14505, 34055, 80450, 191166, 455473, 1086863, 2592817

Offset: 0

Max Alekseyev, May 16 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Cf. A000390, A119266, A119267, A119268, A119269, A119338, A119341, A119342.

a(18)-a(21) from Max Alekseyev, Mar 30 2025

A119341 Number of 6-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 58, 119, 257, 562, 1268, 2910, 6844, 16371, 39910, 98667, 247200, 625559

Offset: 0

Max Alekseyev, May 16 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Cf. A000416, A119266, A119267, A119268, A119269, A119338, A119340, A119342.

a(16)-a(19) from Max Alekseyev, Mar 31 2025

A119342 Number of 7-dimensional partitions of n up to conjugacy.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 28, 58, 120, 259, 568, 1287, 2970, 7036, 17009, 42042, 105848

Offset: 0

Max Alekseyev, May 16 2006

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

Cf. A000427, A119266, A119267, A119268, A119269, A119338, A119340, A119341.

a(15)-a(17) from Max Alekseyev, Apr 02 2025

A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 7, 12, 16, 22, 32, 50, 68, 96, 134, 195, 261, 364, 497, 701, 941, 1288, 1738

Offset: 1

Wouter Meeussen, Mar 19 2025

Permutes the 4 axes of the 4D-Ferrers plot of the solid partitions as 2143.

			z[{{2},{2}}] -> z[{{1,1}},{{1,1}}] -> z[{{2},{2}}] under the 'lapse' operation.

Wouter Meeussen, Mma functions for plane and solid partitions

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

Tr/@Table[Count[solidformBTK[par], arg_z/; Nest[lapse,arg,2]===arg], {n, 20}, {par, IntegerPartitions[n]}]

cp's OEIS Frontend

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

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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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A119269 Table by antidiagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.

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A119338 Table by antidiagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.

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A119267 Number of 4-dimensional partitions of n up to conjugacy.

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A119340 Number of 5-dimensional partitions of n up to conjugacy.

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A119341 Number of 6-dimensional partitions of n up to conjugacy.

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A119342 Number of 7-dimensional partitions of n up to conjugacy.

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A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

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