A061260 -id:A061260 - cp's OEIS frontend

A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 1, 1, 0, 3, 4, 3, 1, 1, 0, 4, 8, 5, 3, 1, 1, 0, 5, 11, 10, 5, 3, 1, 1, 0, 6, 18, 16, 11, 5, 3, 1, 1, 0, 8, 25, 29, 18, 11, 5, 3, 1, 1, 0, 10, 38, 44, 34, 19, 11, 5, 3, 1, 1, 0, 12, 52, 72, 55, 36, 19, 11, 5, 3, 1, 1

Offset: 0

Alois P. Heinz, Apr 14 2017

			T(n,k) is the number of multisets of exactly k partitions of positive integers into distinct parts with total sum of parts equal to n.
T(4,1) = 2: {4}, {31}.
T(4,2) = 3: {3,1}, {21,1}, {2,2}.
T(4,3) = 1: {2,1,1}.
T(4,4) = 1: {1,1,1,1}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,   1;
  0,  2,  3,   1,  1;
  0,  3,  4,   3,  1,  1;
  0,  4,  8,   5,  3,  1,  1;
  0,  5, 11,  10,  5,  3,  1,  1;
  0,  6, 18,  16, 11,  5,  3,  1,  1;
  0,  8, 25,  29, 18, 11,  5,  3,  1, 1;
  0, 10, 38,  44, 34, 19, 11,  5,  3, 1, 1;
  0, 12, 52,  72, 55, 36, 19, 11,  5, 3, 1, 1;
  0, 15, 75, 110, 96, 60, 37, 19, 11, 5, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Index entries for triangles generated by the Multiset Transformation

Columns k=0..10 give: A000007, A000009 (for n>0), A320787, A320788, A320789, A320790, A320791, A320792, A320793, A320794, A320795.
Row sums give A089259.
T(2n,n) give A285230.
Cf. A061260, A360763.

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
       x^j*binomial(g(i)+j-1, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..16);

L[n_] := QPochhammer[x^2]/QPochhammer[x] + O[x]^n;
A[n_] := Module[{c = L[n]}, CoefficientList[#, y]& /@ CoefficientList[ 1/Product[(1 - x^k*y + O[x]^n)^SeriesCoefficient[c, {x, 0, k}], {k, 1, n}], x]];
A[12] // Flatten (* Jean-François Alcover, Jan 19 2020, after Andrew Howroyd *)
g[n_] := g[n] = If[n==0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]*x^j* Binomial[g[i] + j - 1, j], {j, 0, n/i}]]];
T[n_] := CoefficientList[b[n, n] + O[x]^(n+1), x];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
  1
  0  1
  0  2  0
  0  3  2  0
  0  5  4  0  0
  0  7 11  1  0  0
  0 11 20  6  0  0  0
  0 15 40 16  0  0  0  0
  0 22 68 40  3  0  0  0  0
  ...
Row n = 5 counts the following sets of multisets:
  {{5}}          {{1},{4}}        {{1},{2},{1,1}}
  {{1,4}}        {{2},{3}}
  {{2,3}}        {{1},{1,3}}
  {{1,1,3}}      {{1},{2,2}}
  {{1,2,2}}      {{2},{1,2}}
  {{1,1,1,2}}    {{3},{1,1}}
  {{1,1,1,1,1}}  {{1},{1,1,2}}
                 {{1,1},{1,2}}
                 {{2},{1,1,1}}
                 {{1},{1,1,1,1}}
                 {{1,1},{1,1,1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
       combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 30 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
     PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - Andrew Howroyd, Dec 29 2019

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A119442 Triangle read by rows: row n lists number of unordered partitions of n into k parts which are partition numbers (members of A000041).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 7, 2, 1, 7, 11, 7, 2, 1, 11, 26, 19, 7, 2, 1, 15, 40, 38, 19, 7, 2, 1, 22, 83, 78, 54, 19, 7, 2, 1, 30, 120, 168, 102, 54, 19, 7, 2, 1, 42, 223, 301, 244, 134, 54, 19, 7, 2, 1, 56, 320, 557, 471, 292, 134, 54, 19, 7, 2, 1, 77, 566, 1035, 1000, 623, 356, 134, 54

Offset: 0

Alford Arnold, May 19 2006

A060642 describes the ordered case.

Number of twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. - Gus Wiseman, Mar 23 2018

			Triangle begins:
   1
   2   1
   3   2   1
   5   7   2   1
   7  11   7   2   1
  11  26  19   7   2   1
  15  40  38  19   7   2   1
  22  83  78  54  19   7   2   1
  30 120 168 102  54  19   7   2   1
  42 223 301 244 134  54  19   7   2   1
  56 320 557 471 292 134  54  19   7   2   1
The T(5,3) = 7 twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(2)(1), (11)(11)(1). - _Gus Wiseman_, Mar 23 2018

Cf. A000041, A001970, A008284, A036036, A048996, A055887, A061260, A063834, A273873, A281145, A289501, A299200, A299201.

nn=12;
ser=Product[1/(1-PartitionsP[n]x^n y),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n},{y,0,k}],{n,nn},{k,n}] (* Gus Wiseman, Mar 23 2018 *)

G.f.: 1/Product_{k>0} (1-y*A000041(k)*x^k). - Vladeta Jovovic, May 21 2006

More terms and better definition from Vladeta Jovovic, May 21 2006

A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

Original entry on oeis.org

1, 0, 1, 0, 4, 2, 0, 10, 8, 3, 0, 33, 48, 18, 5, 0, 91, 204, 118, 32, 7, 0, 298, 959, 743, 266, 58, 11, 0, 910, 4193, 4334, 1927, 519, 94, 15, 0, 3017, 18947, 25305, 13992, 4407, 966, 154, 22, 0, 9945, 84798, 145033, 97947, 36410, 9023, 1679, 236, 30

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
   1
   0   1
   0   4   2
   0  10   8   3
   0  33  48  18   5
   0  91 204 118  32   7
   0 298 959 743 266  58  11
For example, row n = 3 counts the following multiset partitions:
  {{111}}      {{1}}{{11}}    {{1}}{{1}}{{1}}
  {{112}}      {{1}}{{12}}    {{1}}{{1}}{{2}}
  {{123}}      {{1}}{{23}}    {{1}}{{2}}{{3}}
  {{1}{11}}    {{2}}{{11}}
  {{1}{12}}    {{1}}{{1}{1}}
  {{1}{23}}    {{1}}{{1}{2}}
  {{2}{11}}    {{1}}{{2}{3}}
  {{1}{1}{1}}  {{2}}{{1}{1}}
  {{1}{1}{2}}
  {{1}{2}{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A007716 (for n > 0).
Column k = n is A000041.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
If this is the 3-dimensional version, the 2-dimensional version is A317533.
See A330473 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k,n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(A,k,x)*x^k + O(x*x^n), sExp(A)) ))}
M(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 17 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2023

A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 3, 8, 10, 0, 5, 28, 38, 33, 0, 7, 56, 146, 152, 91, 0, 11, 138, 474, 786, 628, 298, 0, 15, 268, 1388, 3117, 3808, 2486, 910, 0, 22, 570, 3843, 11830, 19147, 18395, 9986, 3017, 0, 30, 1072, 10094, 40438, 87081, 110164, 86388, 39889, 9945

Offset: 0

Gus Wiseman, Dec 20 2019

As an alternative description, T(n,k) is the number of non-isomorphic multisets of nonempty multisets of nonempty multisets with n leaves whose multiset union consists of k multisets.

			Triangle begins:
   1
   0   1
   0   2   4
   0   3   8  10
   0   5  28  38  33
   0   7  56 146 152  91
   0  11 138 474 786 628 298
For example, row n = 3 counts the following multiset partitions:
  {{111}}  {{1}{11}}    {{1}{1}{1}}
  {{112}}  {{1}{12}}    {{1}{1}{2}}
  {{123}}  {{1}{23}}    {{1}{2}{3}}
           {{2}{11}}    {{1}}{{1}{1}}
           {{1}}{{11}}  {{1}}{{1}{2}}
           {{1}}{{12}}  {{1}}{{2}{3}}
           {{1}}{{23}}  {{2}}{{1}{1}}
           {{2}}{{11}}  {{1}}{{1}}{{1}}
                        {{1}}{{1}}{{2}}
                        {{1}}{{2}}{{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..350

Row sums are A318566.
Column k = 1 is A000041 (for n > 0).
Column k = n is A007716.
Partitions of partitions of partitions are A007713.
Twice-factorizations are A050336.
The 2-dimensional version is A317533.
See A330472 for a variation.
Cf. A001055, A050336, A061260, A269134, A292504, A306186, A317791.

\\ See links in A339645 for combinatorial species functions.
ColGf(k, n)={my(A=symGroupSeries(n)); OgfSeries(sCartProd(sExp(A), sSubstOp(polcoef(sExp(A), k, x)*x^k + O(x*x^n), A) ))}
M(n, m=n)={Mat(vector(m+1, k, Col(ColGf(k-1, n), -(n+1))))}
{ my(A=M(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 18 2023

Terms a(36) and beyond from Andrew Howroyd, Jan 18 2023

A356957 Number of set partitions of strict integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 7, 7, 8, 13, 20, 19, 27, 30, 42, 60, 63, 75, 99, 112, 141, 191, 205, 248, 296, 357, 408, 513, 617, 696, 831, 969, 1117, 1337, 1523, 1797, 2171, 2420, 2805, 3265, 3772, 4289, 5013, 5661, 6579, 7679, 8615, 9807, 11335, 12799, 14581

Offset: 0

Gus Wiseman, Sep 13 2022

nonn

			The a(1) = 1 through a(6) = 7 set partitions:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1},{3}}  {{2,3}}    {{1,2,3}}
                {{1},{2}}             {{1},{4}}  {{1},{5}}
                                      {{2},{3}}  {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{1,2},{3}}
                                                 {{1},{2},{3}}

Intervals are counted by A000012, A001227, ranked by A073485.
The initial version is A010054.
For set partitions of {1..n} we have A011782.
The non-strict version is A107742
Not restricting to intervals gives A294617.
More types: A356936, A356937, A356938, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A001970 counts multiset partitions of integer partitions.
A356941 counts multiset partitions of integer partitions w/ gapless blocks.
Cf. A001055, A055887, A061260, A270995, A304969, A349050, A349055.

chQ[y_] := Length[y] <= 1 || Union[Differences[y]] == {1};
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@sps/@Reverse/@Select[IntegerPartitions[n], UnsameQ@@#&],And@@chQ/@#&]],{n,0,15}]

cp's OEIS Frontend

A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

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A301595 Number of thrice-partitions of n.

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A119442 Triangle read by rows: row n lists number of unordered partitions of n into k parts which are partition numbers (members of A000041).

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A330472 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element multisets of nonempty multisets of nonempty multisets (all finite).

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A330473 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of k-element multiset partitions of multisets of size n.

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A356957 Number of set partitions of strict integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1.

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