A323450 -id:A323450

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A089299

Number of square plane partitions of n.

+10
13

1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 41, 57, 78, 108, 146, 202, 274, 375, 509, 690, 929, 1255, 1679, 2246, 2991, 3979, 5266, 6971, 9187, 12104, 15898, 20870, 27322, 35762, 46690, 60927, 79348, 103270, 134138, 174108, 225576, 291990, 377320, 487083

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Number of ways of writing n as a sum p(1,1) + p(1,2) + ... + p(1,k) + p(2,1) + ... + p(2,k) + ... + p(k,1) + ... + p(k,k) for some k so that in the square array {p(i,j)} the numbers are nonincreasing along rows and columns. All the p(i,j) are >= 1.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..4000

FORMULA

G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..2k-1} (1-x^j)^min(j,2k-j). - Franklin T. Adams-Watters, Jun 14 2006

EXAMPLE

a(7) = 5:

7 41 32 31 22

. 11 11 21 21

a(10) = 16 from {{10}}, {{3, 2}, {3, 2}}, {{3, 3}, {2, 2}}, {{3, 3}, {3, 1}}, {{4, 1}, {4, 1}}, {{4, 2}, {2, 2}}, {{4, 2}, {3, 1}}, {{4, 3}, {2, 1}}, {{4, 4}, {1, 1}}, {{5, 1}, {3, 1}}, {{5, 2}, {2, 1}}, {{5, 3}, {1, 1}}, {{6, 1}, {2, 1}}, {{6, 2}, {1, 1}}, {{7, 1}, {1, 1}}, {{2, 1, 1}, {1, 1, 1}, {1, 1, 1}}

From Gus Wiseman, Jan 16 2019: (Start)

The a(10) = 16 square plane partitions:

[ten]

[32] [33] [33] [41] [42] [42] [43] [44] [51] [52] [53] [61] [62] [71]

[32] [22] [31] [41] [22] [31] [21] [11] [31] [21] [11] [21] [11] [11]

[211]

[111]

(End)

MATHEMATICA

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#, {Length[ptn]}]&/@ptn]], And@@OrderedQ/@Transpose[#]&]], {ptn, IntegerPartitions[n]}], {n, 30}] (* Gus Wiseman, Jan 16 2019 *)

CROSSREFS

Cf. A008763, A001970, A089292.

Cf. A000219, A003293, A101509, A319066, A323429, A323433, A323450.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 25 2003

EXTENSIONS

Corrected and extended by Wouter Meeussen, Dec 30 2003

a(21)-a(25) from John W. Layman, Jan 02 2004

More terms from Franklin T. Adams-Watters, Jun 14 2006

Name edited by Gus Wiseman, Jan 16 2019

STATUS

approved

A323438

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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.

LINKS

Table of n, a(n) for n=1..85.

Wikipedia, Young tableau

FORMULA

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

EXAMPLE

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

2 1

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 16 2019

STATUS

approved

A323439

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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.

LINKS

Table of n, a(n) for n=1..87.

Wikipedia, Young tableau

FORMULA

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

EXAMPLE

The a(630) = 8 tableaux:

123 124 1234

24 23 2

12 12 123 124

23 24 2 2

4 3 4 3

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 16 2019

STATUS

approved

A323582

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

For strictly increasing columns, see A100883.

LINKS

Table of n, a(n) for n=0..40.

EXAMPLE

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

2 1

MATHEMATICA

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 19 2019

EXTENSIONS

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

STATUS

approved

A323451

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..50

nLab, Young Diagram.

The Unapologetic Mathematician weblog, Generalized Young Tableaux.

EXAMPLE

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

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}]

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 16 2019

EXTENSIONS

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

STATUS

approved

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