OFFSET
0,4
COMMENTS
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]).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Franklin T. Adams-Watters)
OEIS Wiki, Plane partitions
EXAMPLE
From Gus Wiseman, Nov 15 2018: (Start)
The a(10) = 35 strict plane partitions (A = 10):
A 64 73 82 532 91 541 631 721 4321
.
9 54 63 72 432 8 53 71 431 7 43 52 61 421 6 42 51
1 1 1 1 1 2 2 2 2 3 21 3 3 3 4 31 4
.
7 6 5 43 42 5 41
2 3 4 2 3 3 3
1 1 1 1 1 2 2
.
4
3
2
1
(End)
MAPLE
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
-> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
end:
g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:
a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:
seq(a(n), n=0..60); # Alois P. Heinz, Nov 18 2012
MATHEMATICA
prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
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 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]];
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]]];
g[n_] := g[n] = If[n < 2, 1, (n - 1)*g[n - 2] + g[n - 1]];
a[n_] := With[{bn = b[n, n]}, Sum[bn[[i]]*g[i - 1], {i, 1, Length[bn]}]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Mar 16 2006, Apr 01 2008
STATUS
approved