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

%I A000786 M1020 N0383 #38 Feb 16 2025 08:32:21
%S A000786 1,1,1,2,4,6,11,19,33,55,95,158,267,442,731,1193,1947,3137,5039,8026,
%T A000786 12726,20024,31373,48835,75673,116606,178889,273061,415086,628115,
%U A000786 946723,1421082,2125207,3166152,4700564,6954151,10254486,15071903
%N A000786 Number of inequivalent planar partitions of n, when considering them as 3D objects.
%C A000786 Partitions that are the same when regarded as 3-D objects are counted only once. - _Wouter Meeussen_, May 2006
%D A000786 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
%D A000786 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000786 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000786 Jean-François Alcover, <a href="/A000786/b000786.txt">Table of n, a(n) for n = 0..150</a>
%H A000786 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%H A000786 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MacdonaldsPlanePartitionConjecture.html">Macdonald's Plane Partition Conjecture</a>.
%H A000786 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>.
%F A000786 Equals A000784 + A000785 + A048141 + A048142.
%F A000786 Equals (A048141 + 3*A048140 - A000219 + 2*A048142)/3. - _Wouter Meeussen_, May 2006
%e A000786 From _M. F. Hasler_, Oct 01 2018: (Start)
%e A000786 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.
%e A000786 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.
%e A000786 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.
%e A000786 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)
%t A000786 nmax = 150;
%t A000786 a219[0] = 1;
%t A000786 a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
%t A000786 s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[( nmax+1)/2]}] + O[x]^(nmax+1);
%t A000786 A005987 = CoefficientList[s, x];
%t A000786 a048140[n_] := (a219[n] + A005987[[n+1]])/2;
%t A000786 A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
%t A000786 A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {_, _}][[All, 2]];
%t A000786 a[0] = 1;
%t A000786 a[n_] := (A048141[[n]] + 3 a048140[n] - a219[n] + 2 A048142[[n]])/3;
%t A000786 a /@ Range[0, nmax] (* _Jean-François Alcover_, Dec 28 2019 *)
%Y A000786 Cf. A000784, A000785, A000219, A005987, A048142, A051056-A051061, A096419.
%K A000786 nonn,easy,nice
%O A000786 0,4
%A A000786 _N. J. A. Sloane_
%E A000786 More terms from _Wouter Meeussen_, 1999
%E A000786 Name & links edited and a(0) = 1 added by _M. F. Hasler_, Sep 30 2018