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 1-6 of 6 results.

A323429 Number of rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 58, 81, 124, 169, 257, 345, 501, 684, 968, 1304, 1830, 2452, 3387, 4541, 6188, 8257, 11193, 14865, 19968, 26481, 35341, 46674, 62007, 81611, 107860, 141602, 186292, 243800, 319610, 416984, 544601, 708690, 922472, 1197018, 1553442
Offset: 0

Views

Author

Gus Wiseman, Jan 15 2019

Keywords

Comments

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing.

Examples

			The a(5) = 14 matrices:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Crossrefs

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.
Cf. A323301, A323307, A323430, A323431, A323432, A323435, A323436, A323438.

Programs

  • Mathematica
    Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323430 Number of rectangular plane partitions of n with strictly decreasing rows and columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 12, 16, 22, 27, 36, 44, 57, 72, 89, 110, 139, 170, 210, 261, 318, 390, 478, 581, 705, 860, 1036, 1252, 1511, 1816, 2178, 2618, 3127, 3743, 4471, 5330, 6347, 7564, 8984, 10674, 12669, 15016, 17780, 21050, 24868, 29371, 34655, 40836, 48080
Offset: 0

Views

Author

Gus Wiseman, Jan 15 2019

Keywords

Comments

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are strictly decreasing.

Examples

			The a(8) = 12 matrices:
  [8] [7 1] [6 2] [5 3] [5 2 1] [4 3 1]
.
  [7] [6] [5] [3 2]
  [1] [2] [3] [2 1]
.
  [5] [4]
  [2] [3]
  [1] [1]
The a(10) = 22 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [5 2] [4 2] [4 3]
  [1] [2] [3] [4] [2 1] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Crossrefs

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.
Cf. A323301, A323307, A323429, A323431, A323432, A323435, A323436, A323438.

Programs

  • Mathematica
    Table[Sum[Length[Select[Union[Tuples[Select[IntegerPartitions[#,{k}],UnsameQ@@#&]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323432 Number of semistandard rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 20, 30, 42, 59, 79, 112, 146, 199, 264, 350, 455, 603, 774, 1010, 1297, 1668, 2124, 2724, 3441, 4372, 5513, 6955, 8718, 10960, 13670, 17091, 21264, 26454, 32786, 40667, 50215, 62048, 76435, 94126
Offset: 0

Views

Author

Gus Wiseman, Jan 16 2019

Keywords

Comments

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows are weakly decreasing and the columns are strictly decreasing.

Examples

			The a(6) = 15 matrices:
  [6] [51] [42] [411] [33] [321] [3111] [222] [2211] [21111] [111111]
.
  [5] [4] [22]
  [1] [2] [11]
.
  [3]
  [2]
  [1]

Crossrefs

Cf. A000219, A003293, A047966, A089299, A101509, A114736, A117433, A299968, A319066.
Cf. A323301, A323307, A323429, A323430, A323431, A323435, A323436, A323438.

Programs

  • Mathematica
    Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 20, 24, 31, 38, 48, 59, 72, 86, 106, 125, 150, 180, 213, 250, 296, 347, 407, 477, 555, 645, 751, 869, 1003, 1161, 1334, 1534, 1763, 2018, 2306, 2637, 3002, 3418, 3886, 4409, 4994, 5659, 6390, 7214, 8135, 9160, 10300, 11580, 12990
Offset: 0

Views

Author

Gus Wiseman, Jan 15 2019

Keywords

Examples

			The a(10) = 20 split partitions:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 3]
  [1] [2] [3] [4] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Links

Crossrefs

Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
         b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2019
  • Mathematica
    Table[Sum[Length[Divisors[Length[ptn]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
     If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
     b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.

A323435 Number of rectangular plane partitions of n with no repeated rows or columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 8, 13, 15, 28, 33, 52, 69, 101, 133, 202, 256, 369, 506, 688, 935, 1295, 1736, 2355, 3184, 4284, 5745, 7722, 10281, 13691, 18316, 24168, 32058, 42389, 55915, 73542, 96753, 126709, 166079, 217017, 283258
Offset: 0

Views

Author

Gus Wiseman, Jan 15 2019

Keywords

Comments

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing and with no repeated rows or columns.

Examples

			The a(7) = 13 plane partitions:
  [7] [4 3] [5 2] [6 1] [4 2 1]
.
  [6] [5] [3 2] [4 1] [4] [2 2] [3 1]
  [1] [2] [1 1] [1 1] [3] [2 1] [2 1]
.
  [4]
  [2]
  [1]

Crossrefs

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.
Cf. A323301, A323307, A323429, A323430, A323431, A323432, A323436, A323438.

Programs

  • Mathematica
    Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And[UnsameQ@@#,UnsameQ@@Transpose[#],And@@(OrderedQ[#,GreaterEqual]&/@Transpose[#])]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,20}]

A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Examples

			The a(10) = 25 matrices:
  [10]
.
  [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
  [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
.
  [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
  [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
.
  [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
  [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]

Links

Crossrefs

Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.

Programs

  • 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:
a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019
  • Mathematica
    Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n,{k^2}],UnsameQ@@#&]],{k,n}],{n,20}]
(* Second program: *)
q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
     If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 20 2021 *)

Formula

a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.
Showing 1-6 of 6 results.

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