A323439 -id:A323439 - cp's OEIS frontend

A323436 Number of plane partitions whose parts are the prime indices of n.

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 7, 2, 4, 2, 3, 1, 7, 2, 5, 2, 2, 1, 8, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 4, 3, 2, 4, 1, 7, 5, 2, 1, 8, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

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

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.

			The a(120) = 12 plane partitions:
  32111
.
  311   321   3111   3211
  21    11    2      1
.
  31   32   311   321
  21   11   2     1
  1    1    1     1
.
  31   32
  2    1
  1    1
  1    1
.
  3
  2
  1
  1
  1

Cf. A000085, A000219, A003293, A056239, A112798, A114736, A117433, A138178, A296188, A299968.

Cf. A323300, A323429, A323437, A323438, A323439.

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[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,100}]

A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 5, 2, 2, 2

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a Young diagram with the prime indices of n such that all rows are weakly increasing and all columns are strictly increasing.
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.
Is this a duplicate of A339887? - R. J. Mathar, Feb 03 2021

			The a(60) = 5 tableaux:
  1123
.
  11   112   113
  23   3     2
.
  11
  2
  3

Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.

Cf. A323300, A323429, A323432, A323436, A323438, A323439.

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@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A003293(n).

A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 56, 103, 203, 374, 702, 1262, 2306, 4078, 7242, 12628, 21988, 37756, 64682, 109606, 185082, 309958, 516932, 856221, 1412461, 2316416, 3783552

Offset: 0

Gus Wiseman, Jan 16 2019

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

			The a(4) = 14 generalized Young tableaux:
  4   1 3   2 2   1 1 2   1 1 1 1
.
  1   2   1 1   1 2   1 1   1 1 1
  3   2   2     1     1 1   1
.
  1   1 1
  1   1
  2   1
.
  1
  1
  1
  1
The a(5) = 26 generalized Young tableaux:
  5   1 4   2 3   1 1 3   1 2 2   1 1 1 2   1 1 1 1 1
.
  1   2   1 1   1 3   1 2   1 1   1 1 1   1 1 2   1 1 1   1 1 1 1
  4   3   3     1     2     1 2   2       1       1 1     1
.
  1   1   1 1   1 2   1 1   1 1 1
  1   2   1     1     1 1   1
  3   2   2     1     1     1
.
  1   1 1
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1

nLab, Young Diagram.
The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Cf. A000085, A000219, A003293, A053529, A114736, A138178, A296188, A299968.

Cf. A323436, A323437, A323438, A323439, A323451.

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[Sum[Length[Select[ptnplane[Times@@Prime/@y],And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&]],{y,IntegerPartitions[n]}],{n,10}]

a(16)-a(26) from Seiichi Manyama, Aug 19 2020

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

Gus Wiseman, Jan 16 2019

nonn

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

			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

Seiichi Manyama, Table of n, a(n) for n = 0..50
nLab, Young Diagram.
The Unapologetic Mathematician weblog, Generalized Young Tableaux.

Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.

Cf. A323430, A323436, A323437, A323439, A323450.

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}]

a(21)-a(45) from Seiichi Manyama, Aug 19 2020

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

Gus Wiseman, Jan 17 2019

nonn

			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]

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

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

A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

Original entry on oeis.org

1, 1, 3, 6, 13, 23, 45, 76, 136, 225, 381, 611, 1001, 1570, 2489, 3842, 5948, 9022, 13714, 20501, 30649, 45262, 66721, 97393, 141888, 204993

Offset: 0

Gus Wiseman, Jan 18 2019

			The a(5) = 23 tableaux:
  5   41   32   311   221   2111   11111
.
  1   2   11   21   11   111   111   1111
  4   3   3    2    21   2     11    1
.
  1   1   11   11   111
  1   2   1    11   1
  3   2   2    1    1
.
  1   11
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1

nLab, Young Diagram.

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323581.

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]]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,facs[y],{2}],And@@(GreaterEqual@@@Transpose[PadRight[#]])&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

A323581 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 14, 19, 28, 34, 48, 60, 80, 106, 134, 171, 222, 279, 354, 452, 562, 706, 884, 1100

Offset: 0

Gus Wiseman, Jan 18 2019

			The a(8) = 14 tableaux:
  8   1 7   2 6   3 5   1 2 5   1 3 4
.
  7   6   5   2 5   3 4   2 3
  1   2   3   1     1     1 2
.
  5   4
  2   3
  1   1

nLab, Young Diagram.

Cf. A000085, A000219, A003293, A114736, A138178, A299968, A323436, A323437, A323438, A323439, A323580.

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]}]];
Table[Sum[Length[Select[Reverse/@Sort/@Map[primeMS,sqfacs[y],{2}],And@@Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}]&]],{y,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

cp's OEIS Frontend

A323436 Number of plane partitions whose parts are the prime indices of n.

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A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

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A323450 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are weakly increasing.

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

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

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A323580 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are weakly decreasing and the columns are weakly increasing.

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A323581 Number of ways to fill a Young diagram with positive integers summing to n such that the rows are strictly increasing and the columns are strictly decreasing.

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