A323529 Number of strict square plane partitions of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 55, 69, 79, 95, 109, 129, 145, 169, 189, 217, 241, 273, 301, 339, 371, 413, 451, 499, 541, 595, 643, 703, 757, 823, 925, 999, 1107, 1229, 1387, 1559, 1807, 2071, 2453, 2893, 3451, 4109, 5011

Offset

0,11

Links

Alois P. Heinz, Table of n, a(n) for n = 0..7500

Formula

a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - Alois P. Heinz, Jan 24 2019

Example

The a(12) = 5 strict square plane partitions:
  [12]
.
  [1 2] [1 2] [1 3] [1 4]
  [3 6] [4 5] [2 6] [2 5]
The a(15) = 13 strict square plane partitions:
  [15]
.
  [7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
  [2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]

Maple

h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
         b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 24 2019

Mathematica

Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#, {Length[ptn]}]&/@ptn]], UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]], {ptn, IntegerPartitions[n]}], {n, 30}]
(* Second program: *)
h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0, If[n == 0, If[IntegerQ[ Sqrt[t]], h[Sqrt[t]], 0], b[n-i, Min[n-i, i-1], t+1] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 70] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000219, A008289, A039622, A047966, A089299, A101509, A319066, A323429, A323434, A323522, A323530.

Sequence in context: A161834 A141867 A163646 * A328222 A306276 A001588

Adjacent sequences: A323526 A323527 A323528 * A323530 A323531 A323532

Keyword

nonn

Author

Gus Wiseman, Jan 17 2019

Extensions

More terms from Alois P. Heinz, Jan 24 2019

Status

approved