A299968 -id:A299968 - cp's OEIS frontend

A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 45, 197, 45, 1, 1, 1, 1, 197, 4593, 4593, 197, 1, 1, 1, 1, 903, 126289, 732963, 126289, 903, 1, 1, 1, 1, 4279, 3888343, 155242003, 155242003, 3888343, 4279, 1, 1, 1, 1, 20793, 130016393, 40007492715, 289599115433, 40007492715, 130016393, 20793, 1, 1

Offset: 0

Andrew Howroyd, Sep 16 2024

T(n,k) is the number of normal generalized Young tableaux with all rows and columns strictly increasing whose shape is a rectangle of size n X k (cf. A299968). - Ludovic Schwob, Nov 18 2024

			Array begins:
=====================================================================
n/k | 0 1   2       3           4               5               6 ...
----+----------------------------------------------------------------
  0 | 1 1   1       1           1               1               1 ...
  1 | 1 1   1       1           1               1               1 ...
  2 | 1 1   3      11          45             197             903 ...
  3 | 1 1  11     197        4593          126289         3888343 ...
  4 | 1 1  45    4593      732963       155242003     40007492715 ...
  5 | 1 1 197  126289   155242003    289599115433 723253222084867 ...
  6 | 1 1 903 3888343 40007492715 723253222084867 ...
...
The T(2,3) = 11 matrices are:
  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 4]  [1 2 4]
  [2 3 4]  [2 4 5]  [3 4 5]  [4 5 6]  [2 3 5]  [3 4 5]
.
  [1 2 4]  [1 2 5]  [1 3 4]  [1 3 4]  [1 3 5]
  [3 5 6]  [3 4 6]  [2 4 5]  [2 5 6]  [2 4 6]

Andrew Howroyd, Table of n, a(n) for n = 0..230
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54.

Columns k=1..4 are A000012, A001003, A105124, A374985.
Main diagonal is A374514.
Cf. A060854 (case all values also distinct), A299968.

\\ See PARI link in A374514 for program code.
for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

T(n,k) = T(k,n).

A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly increasing.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 5, 1, 7, 11, 1, 1, 11, 1, 13, 23, 9, 1, 7, 11, 11, 11, 25, 1, 51, 1, 1, 39, 13, 45, 23, 1, 15, 59, 25, 1, 135, 1, 41, 73, 17, 1, 9, 45, 73, 83, 61, 1, 45, 107, 63, 111, 19, 1, 135, 1, 21, 259, 1, 205, 279, 1, 85, 143, 349, 1

Offset: 1

Gus Wiseman, Feb 26 2018

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. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(15) = 11 tableaux:
1 2 3   1 2 4   1 3 4   1 2 5   1 3 5
4 5     3 5     2 5     3 4     2 4
.
1 2 3   1 2 3   1 2 4   1 2 4   1 3 4
2 4     3 4     2 3     3 4     2 4
.
1 2 3
2 3

Cf. A000085, A001221, A005117, A006958, A015128, A056239, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299968, A300118, A300120, A300122.

a[n_]:=If[n===1,1,Sum[a[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Array[a,100]

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}]

A323586 Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

Original entry on oeis.org

1, 1, 2, 5, 8, 16, 30, 53, 89, 158, 265, 443, 735, 1197

Offset: 0

Gus Wiseman, Jan 20 2019

			The a(4) = 8 plane partitions with no repeated rows:
  4   31   22   211   1111
.
  3   21   111
  1   1    1
The a(6) = 30 plane partitions with no repeated columns:
  6   51   42   321
.
  5   4   41   3   31   32   31   22   21   221   211
  1   2   1    3   2    1    11   2    21   1     11
.
  4   3   31   2   21   22   21   111
  1   2   1    2   2    1    11   11
  1   1   1    2   1    1    1    1
.
  3   2   21   11
  1   2   1    11
  1   1   1    1
  1   1   1    1
.
  2   11
  1   1
  1   1
  1   1
  1   1
.
  1
  1
  1
  1
  1
  1

Cf. A000219, A003293 (strict rows), A114736 (strict rows and columns), A117433 (distinct entries), A299968, A319646 (no repeated rows or columns), A323429, A323436 (plane partitions of type), A323580, A323587.

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[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}]

A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 38, 14, 1, 1, 42, 372, 372, 42, 1, 1, 132, 4282, 14606, 4282, 132, 1, 1, 429, 55149

Offset: 1

Ludovic Schwob, Nov 18 2024

A partition of a poset into antichains is said to be proper if it does not contain two antichains A_1 and A_2, with x_1,y_1 in A_1 and x_2,y_2 in A_2, such that x_1y_2.

A proper antichain partition of a poset is endowed with an order relation, which is induced by the order relation of the poset. Let Y be a young diagram, and P the poset of shape Y. The number of linear extensions of P is the number of standard Young tableaux with shape Y. The sum over all proper antichain partitions of P, of the numbers of linear extensions of the induced orders, is equal to the number of normal generalized Young tableaux of shape Y with all rows and columns strictly increasing (cf. A299968).

			Array begins:
=====================================================================
n/k | 1     2      3      4      5      6 ...
----+----------------------------------------------------------------
  1 | 1     1      1      1      1      1 ...
  2 | 1     2      5     14     42    132 ...
  3 | 1     5     38    372   4282  55149 ...
  4 | 1    14    372  14606 ...
  5 | 1    42   4282 ...
  6 | 1   132  55149 ...

Cf. A060854, A299968, A374985.

T(n,k) = T(k,n).

T(n,2) = A000108(n).

cp's OEIS Frontend

A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

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A285175 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns strictly 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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Mathematica

A323586 Number of plane partitions of n with no repeated rows (or, equivalently, no repeated columns).

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Mathematica

A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

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