A000786 - OEIS

A000786

Number of inequivalent planar partitions of n, when considering them as 3D objects.
(Formerly M1020 N0383)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

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.

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).

LINKS

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.

FORMULA

Equals A000784 + A000785 + A048141 + A048142.

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

EXAMPLE

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)

MATHEMATICA

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 *)