A051056 -id:A051056 - cp's OEIS frontend

A048141 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.

1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 4, 0, 4, 5, 0, 5, 7, 1, 6, 9, 1, 6, 11, 1, 8, 15, 2, 10, 20, 3, 10, 25, 4, 12, 33, 7, 14, 40, 9, 15, 48, 12, 18, 60, 17, 20, 74, 23, 22, 89, 30, 26, 108, 40, 30, 130, 51, 33, 157, 66, 37, 187, 85, 42, 222, 108, 47, 262, 136, 54

Offset: 1

Wouter Meeussen

			The plane partition {{2,1},{1}} has C3v symmetry.

Wouter Meeussen, Table of n, a(n) for n = 1..202

Cf. A000219, A000784, A000785, A000786, A005987, A048142, A051056-A051061, A096419.

A048142 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 2, 2, 0, 3, 3, 0, 5, 6, 0, 7, 9, 0, 11, 16, 1, 14, 23, 2, 20, 36, 4, 27, 52, 7, 37, 78, 13, 48, 111, 21, 65, 163, 36, 83, 227, 56, 109, 322, 89, 139, 444, 135, 179, 618, 207, 226, 841, 305, 288, 1151, 453, 361

Offset: 1

Wouter Meeussen

Only one of the pair left-handed, right-handed, is enumerated.

			The plane partitions {{3, 2, 2}, {3, 1}, {1, 1}} and {{3, 2, 2}, {3, 2}, {1, 1}} have C3 symmetry.

Wouter Meeussen, Table of n, a(n) for n = 1..202

Cf. A000219, A000784, A000785, A000786, A005987, A048141, A051056-A051061, A096419.

a(n) = (A096419(n) - A048141(n))/2.

A000786 Number of inequivalent planar partitions of n, when considering them as 3D objects.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 11, 19, 33, 55, 95, 158, 267, 442, 731, 1193, 1947, 3137, 5039, 8026, 12726, 20024, 31373, 48835, 75673, 116606, 178889, 273061, 415086, 628115, 946723, 1421082, 2125207, 3166152, 4700564, 6954151, 10254486, 15071903

Offset: 0

N. J. A. Sloane

Partitions that are the same when regarded as 3-D objects are counted only once. - Wouter Meeussen, May 2006

			From _M. F. Hasler_, Oct 01 2018: (Start)
For n = 2, all three plane partitions  [2], [1 1] and [1; 1] (where ";" means next row) correspond to a 1 X 1 X 2 rectangular cuboid, therefore a(2) = 1.
For n = 3, we have [3] ~ [1 1 1] ~ [1; 1; 1] all corresponding to a 1 X 1 X 3 rectangular cuboid or tower of height 3, and [2 1] ~ [2; 1] ~ [1 1; 1] correspond to an L-shaped object, therefore a(3) = 2.
For n = 4, [4] ~ [1 1 1 1] ~ [1; 1; 1; 1] correspond to the 4-tower; [3 1] ~ [3; 1] ~ [2 1 1] ~ [2; 1; 1] ~ [1 1 1; 1] ~ [1 1; 1; 1] all correspond to the same L-shaped object, [2 2] ~ [2; 2] ~ [1 1; 1 1] represent a "flat" square, and it remains [2, 1; 1], so a(4) = 4.
For n = 5, we again have the tower [5] ~ [1 1 1 1 1] ~ [1; 1; 1; 1; 1], a "narrow L" or 4-tower with one "foot" [4 1] ~ [4; 1] ~ [2 1 1 1] ~ [2; 1; 1; 1] ~ [1 1 1 1; 1] ~ [1 1; 1; 1; 1], a symmetric L-shape [3 1 1] ~ [3; 1; 1] ~ [1 1 1; 1; 1], a 3-tower with 2 feet [3 1; 1] ~ [2 1; 1; 1] ~ [2 1 1; 1], a flat 2+3 shape [3 2] ~ [3; 2] ~ [2 2 1] ~ [2; 2; 1] ~ [1 1 1; 1 1] ~ [1 1; 1 1; 1] and a 2X2 square with a cube on top, [2 1;1 1] ~ [2 2; 1] ~ [2 1; 2]. This yields a(5) = 6 classes. (End)

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 0..150
P. A. MacMahon, Combinatory analysis.
Eric Weisstein's World of Mathematics, Macdonald's Plane Partition Conjecture.
Eric Weisstein's World of Mathematics, Plane Partition.

Cf. A000784, A000785, A000219, A005987, A048142, A051056-A051061, A096419.

nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[( nmax+1)/2]}] + O[x]^(nmax+1);
A005987 = CoefficientList[s, x];
a048140[n_] := (a219[n] + A005987[[n+1]])/2;
A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {, }][[All, 2]];
A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {, }][[All, 2]];
a[0] = 1;
a[n_] := (A048141[[n]] + 3 a048140[n] - a219[n] + 2 A048142[[n]])/3;
a /@ Range[0, nmax] (* Jean-François Alcover, Dec 28 2019 *)

Equals A000784 + A000785 + A048141 + A048142.

Equals (A048141 + 3*A048140 - A000219 + 2*A048142)/3. - Wouter Meeussen, May 2006

More terms from Wouter Meeussen, 1999

Name & links edited and a(0) = 1 added by M. F. Hasler, Sep 30 2018

cp's OEIS Frontend

A048141 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.

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A048142 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.

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A000786 Number of inequivalent planar partitions of n, when considering them as 3D objects.

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