cp's OEIS Frontend

A117433 Number of planar partitions of n with all part sizes distinct.

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