A003293 -id:A003293 - cp's OEIS frontend

A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

1, 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: 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 are weakly increasing and all columns are strictly increasing.
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.
Is this a duplicate of A339887? - R. J. Mathar, Feb 03 2021

			The a(60) = 5 tableaux:
  1123
.
  11   112   113
  23   3     2
.
  11
  2
  3

Cf. A000085, A000219, A003293, A053529, A056239, A112798, A138178, A153452, A296188.

Cf. A323300, A323429, A323432, A323436, 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[primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Length[Select[ptnplane[y],And[And@@Less@@@#,And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])]&]],{y,100}]

Sum_{A056239(n) = k} a(k) = A003293(n).

A005986 Number of column-strict plane partitions of n.

Original entry on oeis.org

1, 2, 5, 11, 23, 45, 87, 160, 290, 512, 889, 1514, 2547, 4218, 6909, 11184, 17926, 28449, 44772, 69862, 108205, 166371, 254107, 385617, 581729, 872535, 1301722, 1932006, 2853530, 4194867, 6139361, 8946742, 12984724, 18771092, 27033892

Offset: 0

N. J. A. Sloane

nonn

Note that the asymptotic formula by Gordon and Houten, cited in Stanley's paper (proposition 20.3, p. 274) is for sequence A003293, not for A005986. In addition in the same paper proposition 20.2 is wrong and Wright's formula is incomplete (for correct version see A000219). - Vaclav Kotesovec, Feb 28 2015

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio.
Richard P. Stanley, Theory and Application of Plane Partitions, II, Studies in Appl. Math. 50 (1971), 259-279. DOI:10.1002/sapm1971503259.

Cf. A003293.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(modp(n, 2)=0, n+2, n+3)/2): seq(a(n), n=0..45);  # Vaclav Kotesovec, Mar 02 2015 after Alois P. Heinz

CoefficientList[ Series[ Product[1/((1 - x^i)*Product[(1 - x^j), {j, 2 i - 1, 40}]), {i, 40}], {x, 0, 40}], x] (* or *)
CoefficientList[ Series[ Product[1/(1 - x^j)^Floor[(j + 3)/2], {j, 40}], {x, 0, 40}], x] (* Robert G. Wilson v, May 12 2014 *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+5-(-1)^k)/4),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

A005986_list(N,x=(O('x^N)+1)*'x)=Vec(prod(k=1,N,1/(1-x^k)^((k+3)\2))) \\ M. F. Hasler, Sep 26 2018

G.f.: 1/Product((1-x^i)*Product(1-x^j,j=2*i-1..infinity),i=1..infinity) or 1/Product((1-x^i)^floor((i+3)/2),i=1..infinity). - Vladeta Jovovic, May 21 2006

a(n) ~ Zeta(3)^(25/72) * exp(1/24 - 25*Pi^4 / (3456*Zeta(3)) + 5*Pi^2*n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3)*n^(2/3) / 2) / (A^(1/2) * 2^(5/4) * 3^(1/2) * Pi * n^(61/72)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

More terms from Vladeta Jovovic, May 21 2006

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}]

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

A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 5, 2, 1, 1, 6, 10, 9, 5, 2, 1, 1, 7, 14, 16, 10, 5, 2, 1, 1, 8, 19, 24, 19, 11, 5, 2, 1, 1, 9, 24, 37, 32, 21, 11, 5, 2, 1, 1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1, 1, 11, 37, 71, 79, 66, 41, 23, 11, 5, 2, 1, 1, 12, 44, 93, 117, 106, 74, 43, 23, 11, 5, 2, 1, 1, 13, 52, 122, 166, 166, 125, 80, 44, 23, 11, 5, 2, 1, 1, 14, 61, 153, 231, 251, 204, 139, 83, 45, 23, 11, 5, 2, 1, 1, 15, 70, 193, 311, 367, 322, 236, 147, 85, 45, 23, 11, 5, 2, 1

Offset: 1

Wouter Meeussen, Aug 11 2013

Row sums equal A003293.

Reverse of rows seem to converge to A005986: 1, 2, 5, 11, 23, 45, 87, 160, ...

			T(6,3) = 7 since the 7 SSYT with sum of entries = 6 and shape any partition of 3 are
114 , 123 , 222 , 11 ,  12  , 13 ,   1
                  4     3     2      2
                                     3
Triangle starts:
1;
1,  1;
1,  2,  1;
1,  3,  2,  1;
1,  4,  4,  2,  1;
1,  5,  7,  5,  2,  1;
1,  6, 10,  9,  5,  2,  1;
1,  7, 14, 16, 10,  5,  2,  1;
1,  8, 19, 24, 19, 11,  5,  2, 1;
1,  9, 24, 37, 32, 21, 11,  5, 2, 1;
1, 10, 30, 51, 52, 38, 22, 11, 5, 2, 1;

Cf. A003293, A005986.

hooklength[(par_)?PartitionQ]:=Table[Count[par,q_ /; q>=j] +1-i +par[[i]] -j, {i,Length[par]}, {j,par[[i]]} ];
Table[Tr[(SeriesCoefficient[q^(#1 . Range[Length[#1]])/Times @@ (1-q^#1&) /@ Flatten[hooklength[#1]],{q,0,w}]&) /@ Partitions[n]],{w,24},{n,w}]

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

A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 50, 69, 125, 177, 301, 440, 717, 1055, 1675, 2471, 3835, 5660, 8627, 12697, 19095, 27978, 41581, 60650, 89244, 129490, 188925, 272676, 394809, 566882, 815191, 1164510, 1664295, 2365698, 3361844, 4756030, 6723280, 9468138, 13319299

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

First differs from A304967 at a(10) = 50, A304967(10) = 49.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
Also the number of positive integer matrices with only two columns and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{1222}}    {{12222}}    {{112222}}      {{1122222}}
                   {{12}{12}}  {{12}{122}}  {{122222}}      {{1222222}}
                                            {{112}{122}}    {{112}{1222}}
                                            {{12}{1122}}    {{12}{11222}}
                                            {{12}{1222}}    {{12}{12222}}
                                            {{122}{122}}    {{122}{1122}}
                                            {{12}{12}{12}}  {{122}{1222}}
                                                            {{12}{12}{122}}
Inequivalent representatives of the a(8) = 20 matrices:
  [4 4] [3 5] [2 6] [1 7]
.
  [1 1] [1 1] [1 1] [2 1] [2 1] [1 2] [1 2] [3 1] [2 2] [2 2] [1 3]
  [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]
.
  [1 1] [1 1] [1 1] [1 1]
  [1 1] [1 1] [2 1] [1 2]
  [2 2] [1 3] [1 2] [1 2]
.
  [1 1]
  [1 1]
  [1 1]
  [1 1]

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

Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1,(EulerT(vector(n, k, k-1)) + EulerT(vector(n, k, if(k%2, 0, (k+2)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

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}]

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

Original entry on oeis.org

1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.

			The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
  4   31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
  31   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, 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[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.

A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

Original entry on oeis.org

1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

			The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
  31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The same for a(5) = 21:
  41   32   311   221   2111
.
  4   3   31   21   22   21   211   111   1111
  1   2   1    2    1    11   1     11    1
.
  3   2   21   11   111
  1   2   1    11   1
  1   1   1    1    1
.
  2   11
  1   1
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, 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[GCD@@Length/@Split[#]==1,GCD@@Length/@Split[Transpose[PadRight[#]]]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the second is A323584.

cp's OEIS Frontend

A323437 Number of semistandard Young tableaux whose entries are the prime indices of n.

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A005986 Number of column-strict plane partitions of n.

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A323431 Number of strict rectangular plane partitions 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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A228125 Triangle read by rows: T(n,k) = number of semistandard Young tableaux with sum of entries equal to n and shape of tableau a partition of k.

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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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A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

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

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