

A000786


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


10



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
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OFFSET

0,4


COMMENTS

Partitions that are the same when regarded as 3D 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

Table of n, a(n) for n=0..37.
P. A. MacMahon, Combinatory analysis.
E. W. Weisstein, Macdonald's Plane Partition Conjecture, on MathWorldA Wolfram Web Resource.
E. W. Weisstein, Plane Partition, on MathWorldA Wolfram Web Resource.


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 Lshaped object, therefore a(3) = 2.
For n = 4, [4] ~ [1 1 1 1] ~ [1; 1; 1; 1] correspond to the 4tower; [3 1] ~ [3; 1] ~ [2 1 1] ~ [2; 1; 1] ~ [1 1 1; 1] ~ [1 1; 1; 1] all correspond to the same Lshaped 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 4tower 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 Lshape [3 1 1] ~ [3; 1; 1] ~ [1 1 1; 1; 1], a 3tower 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)


CROSSREFS

Cf. A000784, A000785, A000219, A005987, A048142, A051056A051061, A096419.
Sequence in context: A136424 A116732 A048239 * A289080 A000694 A164137
Adjacent sequences: A000783 A000784 A000785 * A000787 A000788 A000789


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Wouter Meeussen, 1999
Name & links edited and a(0) = 1 added by M. F. Hasler, Sep 30 2018


STATUS

approved



