cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A323529 Number of strict square plane partitions of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Examples

			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]

Links

Crossrefs

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

Programs

  • 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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Jan 24 2019

A323530 Number of square plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 25, 30, 39, 46, 58, 67, 82, 94, 112, 127, 149, 168, 194, 218, 251, 282, 324, 368, 425, 489, 573, 670, 797, 952, 1148, 1392, 1703, 2086, 2568, 3168, 3908, 4823, 5947, 7318, 8986, 11012, 13443, 16371, 19866
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Examples

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

Crossrefs

Cf. A000219, A089299, A306320, A323429, A323434, A323439, A323522, A323529, A323530, A323531.

Programs

  • Mathematica
    Table[Sum[Length[Select[Tuples[IntegerPartitions[#,{Length[ptn]}]&/@ptn],And@@Greater@@@#&&And@@Greater@@@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]

A339445 Number of partitions of n into squares such that the number of parts is a square.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 5, 2, 4, 6, 1, 4, 6, 3, 7, 6, 4, 10, 6, 4, 10, 9, 6, 11, 10, 8, 10, 10, 11, 14, 16, 11, 15, 19, 10, 17, 22, 13, 24, 23, 16, 28, 21, 18, 33, 30, 24, 33, 33, 29, 33, 37, 33, 43, 45, 35, 49
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2020

Keywords

Examples

			                                    [1 1 1]
                          [1 4]     [1 1 1]
a(23) = 2 because we have [9 9] and [4 4 9].

Links

Crossrefs

Cf. A000290, A001156, A085755, A089333, A103198, A323522, A339444.

Programs

  • Maple
    g:= proc(n, k, m)
  # number of partitions of n into k parts which are squares > m^2
   option remember; local r;
  if k = 0 then if n = 0 then return 1 else return 0 fi fi;
  if n < k*(m+1)^2 then return 0 fi;
  add(procname(n-r*(m+1)^2, k-r, m+1), r =max(0, ceil((k*(m+2)^2-n)/(2*m+3))) .. k)
end proc:
f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Oct 26 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.