A114736 -id:A114736 - cp's OEIS frontend

A323432 Number of semistandard rectangular plane partitions of n.

1, 1, 2, 4, 6, 9, 15, 20, 30, 42, 59, 79, 112, 146, 199, 264, 350, 455, 603, 774, 1010, 1297, 1668, 2124, 2724, 3441, 4372, 5513, 6955, 8718, 10960, 13670, 17091, 21264, 26454, 32786, 40667, 50215, 62048, 76435, 94126

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows are weakly decreasing and the columns are strictly decreasing.

			The a(6) = 15 matrices:
  [6] [51] [42] [411] [33] [321] [3111] [222] [2211] [21111] [111111]
.
  [5] [4] [22]
  [1] [2] [11]
.
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323431, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 16 2019

nonn

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(630) = 8 tableaux:
  123   124   1234
  24    23    2
.
  12   12   123   124
  23   24   2     2
  4    3    4     3
.
  12
  2
  3
  4

Wikipedia, Young tableau

Cf. A000085, A000219, A056239, A112798, A114736, A117433, A299968, A323300, A323430, A323436, A323438, A323450, 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[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(Less@@@DeleteCases[Transpose[PadRight[#]],0,{2}]),And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

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

A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 13, 15, 23, 33, 49, 59, 83, 101, 133, 281, 321, 477, 655, 941, 1249, 1795, 2241, 3039, 3867, 5047, 6257, 8063, 11459, 13891, 18165, 23149, 29975, 37885, 49197, 61829, 89877, 109165, 145673, 185671, 246131, 310325, 408799, 514485, 668017, 871383

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with all different entries.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:
  {{111}}    {{1111}}    {{11111}}    {{111111}}      {{1111111}}
  {{122}}    {{1222}}    {{11222}}    {{112222}}      {{1112222}}
  {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}      {{1122222}}
                         {{1}{1111}}  {{122333}}      {{1222222}}
                         {{11}{111}}  {{1}{11111}}    {{1223333}}
                                      {{11}{1111}}    {{1}{111111}}
                                      {{1}{11222}}    {{11}{11111}}
                                      {{11}{1222}}    {{111}{1111}}
                                      {{112}{222}}    {{1}{112222}}
                                      {{122}{222}}    {{11}{12222}}
                                      {{2}{11222}}    {{112}{2222}}
                                      {{22}{1222}}    {{122}{2222}}
                                      {{1}{11}{111}}  {{2}{112222}}
                                                      {{22}{12222}}
                                                      {{1}{11}{1111}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000219, A007716, A008289, A059201, A114736, A117433, A120733, A121860, A321653, A321659, A321660, A321661.

(* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];
(* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];
a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];
a /@ Range[0, 45] (* Jean-François Alcover, Sep 14 2019 *)

\\ here b(n) is A121860(n).
b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A323431 Number of strict rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 21, 25, 33, 41, 53, 65, 81, 97, 121, 143, 173, 215, 255, 305, 367, 441, 527, 637, 751, 899, 1067, 1269, 1491, 1775, 2071, 2439, 2875, 3357, 3911, 4577, 5309, 6177, 7171, 8305, 9609, 11151

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of a strict integer partition of n so that the rows and columns are strictly decreasing.

			The a(10) = 21 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 2] [4 3]
  [1] [2] [3] [4] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299 A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A321654 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with distinct row sums and distinct column sums.

Original entry on oeis.org

1, 1, 1, 13, 13, 45, 681, 885, 2805, 8301, 237213

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(3) = 13 matrices:
  [3] [2 1] [1 2]
.
  [2] [2 0] [1 1] [1 1] [1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1]

Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321646, A321652.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Total/@prs2mat[#],UnsameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

A321659 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.

Original entry on oeis.org

1, 1, 1, 9, 9, 17, 161, 169, 313, 465, 5313, 5465, 10457, 15313, 25009, 271929, 286329, 537953, 799121, 1297369, 1805161, 20532897, 21292017, 40508297, 59738825, 97431073, 135137569, 209525865, 2089381929, 2200470833, 4135252289, 6124698121, 9937836505

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(5) = 17 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [4 0] [3] [3 0] [2] [2 0] [1] [1 0] [0 4] [0 3] [0 2] [0 1]
  [1] [0 1] [2] [0 2] [3] [0 3] [4] [0 4] [1 0] [2 0] [3 0] [4 0]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000219, A007716, A008289, A101370, A114736, A117433, A120733, A321645, A321653, A321655, A321660, A321661, A321662.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@DeleteCases[Join@@prs2mat[#],0]]&]],{n,5}]

\\ here b(n) is A101370(n).
b(n)={sum(m=0, n, sum(k=0, m, stirling(m,k,2)*k!)^2*polcoef(log(1+x+O(x*x^n))^m, n)/m!)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*i!*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} A101370(k)*k!*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 45, 49, 85, 125, 233, 273, 417, 529, 745, 2573, 2861, 4761, 6837, 10489, 14317, 22637, 28289, 40041, 52041, 70177, 88561, 117605, 234773, 274761, 407469, 553681, 792613, 1052525, 1493033, 1959009, 3135537, 3904129, 5475673, 7173725, 9853325

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(5) = 9 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [3] [2] [1]
  [1] [2] [3] [4]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000005, A000219, A007716, A008289, A114736, A117433, A120733, A321645, A321653, A321655, A321659, A321661, A321662.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Join@@prs2mat[#]]&]],{n,5}]

seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, (n-1)!*numdiv(n-1) + n!*(numdiv(n) - 2)))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} (k!*A000005(k) + (k+1)!*(A000005(k+1) - 2))*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

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

A321661 Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 22, 25, 40, 58, 186, 204, 347, 478, 734, 2033, 2402, 3814, 5464, 8142, 11058, 30142, 34437, 55940, 77794, 116954, 156465, 229462, 533612, 640544, 994922, 1397896, 2048316, 2778750, 3987432, 5292293, 11921070, 14076550, 21802928, 29917842, 44080285

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, whose nonzero entries are all distinct.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 22 multiset partitions:
  {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}    {{111111}}
                 {{122}}    {{1222}}    {{11222}}    {{112222}}
                 {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}
                 {{1}{22}}  {{1}{222}}  {{1}{1111}}  {{122333}}
                                        {{11}{111}}  {{1}{11111}}
                                        {{11}{222}}  {{11}{1111}}
                                        {{1}{2222}}  {{1}{11222}}
                                                     {{11}{1222}}
                                                     {{11}{2222}}
                                                     {{112}{222}}
                                                     {{11}{2333}}
                                                     {{1}{22222}}
                                                     {{122}{222}}
                                                     {{1}{22333}}
                                                     {{122}{333}}
                                                     {{2}{11222}}
                                                     {{22}{1222}}
                                                     {{1}{11}{111}}
                                                     {{1}{11}{222}}
                                                     {{1}{22}{222}}
                                                     {{1}{22}{333}}
                                                     {{2}{11}{222}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000219, A007716, A008289, A059201, A059849, A114736, A117433, A120733, A321653, A321659, A321660, A321662.

\\ here b(n) is A059849(n).
b(n)={sum(k=0, n, stirling(n,k,1)*sum(i=0, k, stirling(k,i,2))^2)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p,i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} A059849(k)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A323435 Number of rectangular plane partitions of n with no repeated rows or columns.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 8, 13, 15, 28, 33, 52, 69, 101, 133, 202, 256, 369, 506, 688, 935, 1295, 1736, 2355, 3184, 4284, 5745, 7722, 10281, 13691, 18316, 24168, 32058, 42389, 55915, 73542, 96753, 126709, 166079, 217017, 283258

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows and columns are weakly decreasing and with no repeated rows or columns.

			The a(7) = 13 plane partitions:
  [7] [4 3] [5 2] [6 1] [4 2 1]
.
  [6] [5] [3 2] [4 1] [4] [2 2] [3 1]
  [1] [2] [1 1] [1 1] [3] [2 1] [2 1]
.
  [4]
  [2]
  [1]

Cf. A000219, A003293, A047966, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323431, A323432, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And[UnsameQ@@#,UnsameQ@@Transpose[#],And@@(OrderedQ[#,GreaterEqual]&/@Transpose[#])]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,20}]

cp's OEIS Frontend

A323432 Number of semistandard rectangular plane partitions of n.

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A323439 Number of ways to fill a Young diagram with the prime indices of n such that all rows and columns are strictly increasing.

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A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries.

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A323431 Number of strict rectangular plane partitions of n.

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A321654 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with distinct row sums and distinct column sums.

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A321659 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.

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A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

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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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