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%I #26 Dec 05 2023 08:25:04
%S 1,1,1,3,3,5,9,11,15,21,35,41,59,75,103,149,187,243,321,413,527,735,
%T 895,1165,1467,1885,2335,2997,3853,4765,5977,7473,9269,11531,14255,
%U 17537,22201,26897,33233,40613,50027,60637,74459,89963,109751,134407,162117,195859
%N Number of planar partitions of n with all part sizes distinct.
%C Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).
%H Alois P. Heinz, <a href="/A117433/b117433.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Franklin T. Adams-Watters)
%H OEIS Wiki, <a href="https://oeis.org/wiki/Plane_partitions">Plane partitions</a>
%F a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} A000085(k)*A008289(n,k).
%e From _Gus Wiseman_, Nov 15 2018: (Start)
%e The a(10) = 35 strict plane partitions (A = 10):
%e A 64 73 82 532 91 541 631 721 4321
%e .
%e 9 54 63 72 432 8 53 71 431 7 43 52 61 421 6 42 51
%e 1 1 1 1 1 2 2 2 2 3 21 3 3 3 4 31 4
%e .
%e 7 6 5 43 42 5 41
%e 2 3 4 2 3 3 3
%e 1 1 1 1 1 2 2
%e .
%e 4
%e 3
%e 2
%e 1
%e (End)
%p b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
%p -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
%p end:
%p g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:
%p a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Nov 18 2012
%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@DeleteCases[Join@@prs2mat[#],0],And@@(OrderedQ[#,Greater]&/@prs2mat[#]),And@@(OrderedQ[#,Greater]&/@Transpose[prs2mat[#]])]&]],{n,5}] (* _Gus Wiseman_, Nov 15 2018 *)
%t zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]];
%t b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, zip[Plus, b[n, i - 1], If[i > n, {}, Join[{0}, b[n - i, i - 1]]], 0]]];
%t g[n_] := g[n] = If[n < 2, 1, (n - 1)*g[n - 2] + g[n - 1]];
%t a[n_] := With[{bn = b[n, n]}, Sum[bn[[i]]*g[i - 1], {i, 1, Length[bn]}]];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 05 2023, after _Alois P. Heinz_ *)
%Y Cf. A000219, A072706, A117434, A000009.
%Y Cf. A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321652, A321653, A321655, A321659, A321660.
%K nonn
%O 0,4
%A _Franklin T. Adams-Watters_, Mar 16 2006, Apr 01 2008
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