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 1-7 of 7 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}]

A323438 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are weakly increasing.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 3, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 4, 1, 4, 4, 2, 1, 12, 2, 3, 2, 4, 1, 5, 2, 7, 2, 2, 1, 10, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 13, 1, 2, 3, 4, 2, 4, 1, 12, 5, 2, 1, 10, 2
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(96) = 19 tableaux:
  111112
.
  111   1111   1112   11111   11112
  112   12     11     2       1
.
  11   111   111   112   1111   1112
  11   11    12    11    1      1
  12   2     1     1     2      1
.
  11   11   111   112
  11   12   1     1
  1    1    1     1
  2    1    2     1
.
  11   12
  1    1
  1    1
  1    1
  2    1
.
  1
  1
  1
  1
  1
  2

Links

Crossrefs

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A299968, A323300, A323430, A323436, A323438, A323450.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@LessEqual@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Formula

Sum_{A056239(n) = k} a(k) = A323450(n).

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}]

A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 4, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 4, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 4, 1, 0, 2, 4, 1, 0, 1, 2, 1, 0, 2, 4, 1, 0, 0, 2, 1, 0, 2, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(630) = 8 tableaux:
  123   124   1234
  24    23    2
.
  12   12   123   124
  23   24   2     2
  4    3    4     3
.
  12
  2
  3
  4

Links

Crossrefs

Cf. A000085, A000219, A056239, A112798, A114736, A117433, A299968, A323300, A323430, A323436, A323438, A323450, A323451.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(Less@@@DeleteCases[Transpose[PadRight[#]],0,{2}]),And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Formula

Sum_{A056239(n) = k} a(k) = A323451(n).

A323431 Number of strict rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 21, 25, 33, 41, 53, 65, 81, 97, 121, 143, 173, 215, 255, 305, 367, 441, 527, 637, 751, 899, 1067, 1269, 1491, 1775, 2071, 2439, 2875, 3357, 3911, 4577, 5309, 6177, 7171, 8305, 9609, 11151
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 a strict integer partition of n so that the rows and columns are strictly decreasing.

Examples

			The a(10) = 21 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] [4 2] [4 3]
  [1] [2] [3] [4] [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, A089299 A101509, A114736, A117433, A299968, A319066.
Cf. A323301, A323307, A323429, A323430, A323432, A323435, A323436, A323438.

Programs

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

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}]

A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 9, 12, 19, 27, 39, 54, 79, 107, 150, 209, 282, 387, 525, 707, 949, 1272, 1688, 2244, 2968, 3902, 5125, 6712, 8752, 11383, 14780, 19109, 24671, 31768, 40791, 52280, 66860, 85296, 108621, 138054, 175085, 221676, 280161, 353414, 445098, 559661
Offset: 0

Views

Author

Gus Wiseman, Jan 16 2019

Keywords

Comments

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

Examples

			The a(8) = 19 generalized Young tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  1   2   3   1 2   1 5   1 3   1 4   2 3   1 2   1 2 3
  7   6   5   5     2     4     3     3     2 3   2
.
  1   1   1 2
  2   3   2
  5   4   3

Links

Crossrefs

Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.
Cf. A323430, A323436, A323437, A323439, A323450.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
ptnplane[n_]:=Union[Map[primeMS,Join@@Permutations/@sqfacs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]],0,{2}])&]],{y,IntegerPartitions[n]}],{n,10}]

Extensions

a(21)-a(45) from Seiichi Manyama, Aug 19 2020
Showing 1-7 of 7 results.

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

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