A323437 -id:A323437 - 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}]

A323582 Number of generalized Young tableaux with constant rows, weakly increasing columns, and entries summing to n.

Original entry on oeis.org

1, 1, 3, 5, 11, 16, 33, 47, 85, 126, 208, 299, 486, 685, 1050, 1496, 2221, 3097, 4523, 6239, 8901, 12219, 17093, 23202, 32120, 43200, 58899, 78761, 106210, 140786, 188192, 247689, 327965, 429183, 563592, 732730, 955851, 1235370, 1600205, 2057743, 2649254

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

For strictly increasing columns, see A100883.

			The a(5) = 16 tableaux:
  5   1 1 1 1 1
.
  1   2    1 1   1 1 1   1 1 1   1 1 1 1
  4   3    3     2       1 1     1
.
  1   1    1 1   1 1     1 1 1
  1   2    1     1 1     1
  3   2    2     1       1
.
  1   1 1
  1   1
  1   1
  2   1
.
  1
  1
  1
  1
  1

Cf. A000085, A000219, A003293, A006951, A100883, A138178, A279784, A299968, A323432, A323436, A323437, A323438, A323450.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[ptn],And@@SameQ@@@#&&GreaterEqual@@Length/@#&]],{ptn,Sort/@IntegerPartitions[n]}],{n,10}]

a(21)-a(40) from Seiichi Manyama, Aug 20 2020

A339887 Number of factorizations of n into primes or squarefree semiprimes.

Original entry on oeis.org

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: 1

Gus Wiseman, Dec 22 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
Conjecture: also the number of semistandard Young tableaux whose entries are the prime indices of n (A323437).
Is this a duplicate of A323437? - R. J. Mathar, Jan 05 2021

			The a(n) factorizations for n = 36, 60, 180, 360, 420, 840:
  6*6       6*10      5*6*6       6*6*10        2*6*35      6*10*14
  2*3*6     2*5*6     2*6*15      2*5*6*6       5*6*14      2*2*6*35
  2*2*3*3   2*2*15    3*6*10      2*2*6*15      6*7*10      2*5*6*14
            2*3*10    2*3*5*6     2*3*6*10      2*10*21     2*6*7*10
            2*2*3*5   2*2*3*15    2*2*3*5*6     2*14*15     2*2*10*21
                      2*3*3*10    2*2*2*3*15    2*5*6*7     2*2*14*15
                      2*2*3*3*5   2*2*3*3*10    3*10*14     2*2*5*6*7
                                  2*2*2*3*3*5   2*2*3*35    2*3*10*14
                                                2*2*5*21    2*2*2*3*35
                                                2*2*7*15    2*2*2*5*21
                                                2*3*5*14    2*2*2*7*15
                                                2*3*7*10    2*2*3*5*14
                                                2*2*3*5*7   2*2*3*7*10
                                                            2*2*2*3*5*7

Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
Only allowing only primes gives A008966.
Not allowing primes gives A320656.
Unlabeled multiset partitions of this type are counted by A320663/A339888.
Allowing squares of primes gives A320732.
The strict version is A339742.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A000070, A000961, A001221, A096373, A320893, A338914, A339740, A339741, A339841, A339846.

sqpe[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqpe[n/d],Min@@#>=d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[sqpe[n]],{n,100}]

a(A002110(n)) = A000085(n), and in general if n is a product of k distinct primes, a(n) = A000085(k).

a(n) = Sum_{d|n} A320656(n/d), so A320656 is the Moebius transform of this sequence.

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

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}]

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

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A323582 Number of generalized Young tableaux with constant rows, weakly increasing columns, and entries summing to n.

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A339887 Number of factorizations of n into primes or squarefree semiprimes.

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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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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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