A114736 -id:A114736 - cp's OEIS frontend

A323657 Number of strict solid partitions of n.

1, 1, 1, 4, 4, 7, 16, 19, 28, 40, 82, 94, 145, 190, 274, 463, 580, 802, 1096, 1486, 1948, 3148, 3811, 5314, 6922, 9394, 11971, 16156, 23044, 28966, 38368, 50002, 65116, 83872, 108706, 137917, 192070, 236242, 308698, 390772, 506935, 633982, 817324, 1018090

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any number of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.

			The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))       ((5))       ((6))
                ((21))      ((31))      ((32))      ((42))
                ((2)(1))    ((3)(1))    ((41))      ((51))
                ((2))((1))  ((3))((1))  ((3)(2))    ((321))
                                        ((4)(1))    ((4)(2))
                                        ((3))((2))  ((5)(1))
                                        ((4))((1))  ((31)(2))
                                                    ((32)(1))
                                                    ((4))((2))
                                                    ((5))((1))
                                                    ((31))((2))
                                                    ((3)(2)(1))
                                                    ((32))((1))
                                                    ((3)(1))((2))
                                                    ((3)(2))((1))
                                                    ((3))((2))((1))

Alois P. Heinz, Table of n, a(n) for n = 0..740 (first 401 terms from John Tyler Rascoe)

Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.

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}]];
strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]

a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - John Tyler Rascoe, Dec 19 2024

a(21) onwards from John Tyler Rascoe, Dec 19 2024

A002974 Number of restricted solid partitions of n.

Original entry on oeis.org

1, 1, 4, 7, 11, 20, 35, 59, 99, 165, 270, 443, 723, 1161, 1861, 2961, 4654, 7279, 11317, 17476, 26879, 41132, 62601, 94878, 143172, 215115, 321995, 480216, 713655, 1057192

Offset: 1

N. J. A. Sloane

Definition, based on Math. Review MR0297583: By a solid partition of n is meant a 3-dimensional arrangement of positive integers N(x,y,z) satisfying the conditions (i) the integer N(x,y,z) is located at the point with Cartesian coordinates (x,y,z); N(x,y,z) is defined only for certain integers x,y,z >= 0, and (ii) if N(x,y,z) is defined and 0 <= x' <= x, 0 <= y' <= y, 0 <= z' <= z then N(x,y,z) is defined and N(x',y',z') <= N(x,y,z). A solid partition is said to correspond to an (ordinary) partition of n=n_1+n_2+...+n_t, n_k>0, if there is a one-to-one correspondence between the summands n_k and the points (x_k,y_k,z_k) for which N is defined so that n_k=N(x_k,y_k,z_k). Finally, a restricted solid partition is a solid partition such that x'<=x, y'<=y, z'<=z and N(x',y',z')=N(x,y,z) implies x'=x, y'=y, z'=z.

Alternatively, a restricted solid partition is an infinite three-dimensional array of nonnegative integers summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros. - Gus Wiseman, Jan 22 2019

			From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(6) = 20 restricted solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))  ((3))       ((4))          ((5))           ((6))
                ((21))      ((31))         ((32))          ((42))
                ((2)(1))    ((3)(1))       ((41))          ((51))
                ((2))((1))  ((21)(1))      ((3)(2))        ((321))
                            ((3))((1))     ((4)(1))        ((4)(2))
                            ((21))((1))    ((31)(1))       ((5)(1))
                            ((2)(1))((1))  ((3))((2))      ((31)(2))
                                           ((4))((1))      ((32)(1))
                                           ((31))((1))     ((41)(1))
                                           ((3)(1))((1))   ((4))((2))
                                           ((21)(1))((1))  ((5))((1))
                                                           ((31))((2))
                                                           ((3)(2)(1))
                                                           ((32))((1))
                                                           ((41))((1))
                                                           ((3)(1))((2))
                                                           ((3)(2))((1))
                                                           ((4)(1))((1))
                                                           ((31)(1))((1))
                                                           ((3))((2))((1))
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Gupta, Restricted solid partitions, J. Combin. Theory, A 13 (1972), 140-144.

Cf. A000219, A000293 (solid partitions), A000334, A001970, A114736 (restricted plane partitions), A117433 (strict plane partitions), A321662, A323657 (strict solid partitions).

srcplptns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And[And@@(GreaterEqual@@@Transpose[PadRight[#]]),And@@Greater@@@#,And@@(Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}])]&],{ptn,IntegerPartitions[n]}];
srcsolids[n_]:=Join@@Table[Select[Tuples[srcplptns/@y],And[And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)]),And@@(Greater@@@DeleteCases[Transpose[Join@@@(PadRight[#,{n,n}]&/@#)],0,{2}])]&],{y,IntegerPartitions[n]}]
Table[Length[srcsolids[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from Sean A. Irvine, Dec 15 2014

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}]

cp's OEIS Frontend

A323657 Number of strict solid partitions of n.

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A002974 Number of restricted solid partitions of n.

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

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