A323530 -id:A323530 - cp's OEIS frontend

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

Gus Wiseman, Jan 17 2019

nonn

			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]

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

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

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

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 *)

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

More terms from Alois P. Heinz, Jan 24 2019

A306320 Number of square plane partitions of n with distinct row sums and distinct column sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 5, 5, 10, 11, 18, 21, 31, 37, 56, 70, 97, 134, 180, 247, 343, 462, 623, 850, 1128, 1509, 2004, 2649, 3467, 4590, 5958, 7814, 10161, 13287, 17208, 22495, 29129, 37997, 49229, 64098, 82940, 107868, 139390, 180737, 233214, 301527, 388018, 500058

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

			The a(12) = 21 square plane partitions with distinct row sums and distinct column sums:
[twelve]
.
[64][73][82][91][54][63][72][81][44][53][53][62][62][71][43][43][52][52][61]
[11][11][11][11][21][21][21][21][31][22][31][22][31][31][32][41][32][41][41]
.
[221]
[211]
[111]

Cf. A000219, A089299 (square plane partitions), A101509, A271619, A279785, A306318, A323429, A323529, A323530, A323531.

Table[Sum[Length[Select[Union[Reverse/@Sort/@Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Total/@#&&UnsameQ@@Total/@If[#=={},{},Transpose[#]]&&And@@OrderedQ/@Reverse/@If[#=={},{},Transpose[#]]&]],{ptn,IntegerPartitions[n]}],{n,0,20}]

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A323529 Number of strict square plane partitions of n.

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