A320330 -id:A320330 - cp's OEIS frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

1, 1, 1, 3, 4, 7, 12, 19, 30, 49, 77, 119, 186, 286, 438, 670, 1014, 1528, 2300, 3437, 5119, 7603, 11241, 16564, 24343, 35650, 52058, 75820, 110115, 159510, 230522, 332324, 477994, 686044, 982519, 1404243, 2003063, 2851720, 4052429, 5748440, 8140007, 11507125

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

Number of partitions of n into distinct parts with one level of parentheses. Each "part" in parentheses is distinct from all others at the same level. Thus (2+1)+(1) is allowed but (2)+(1+1) and (2+1+1) are not.

			4=(4)=(3)+(1)=(3+1)=(2+1)+(1).
From _Gus Wiseman_, Oct 11 2018: (Start)
a(n) is the number of set systems (sets of sets) whose multiset union is an integer partition of n. For example, the a(1) = 1 through a(6) = 12 set systems are:
  {{1}}  {{2}}  {{3}}      {{4}}        {{5}}        {{6}}
                {{1,2}}    {{1,3}}      {{1,4}}      {{1,5}}
                {{1},{2}}  {{1},{3}}    {{2,3}}      {{2,4}}
                           {{1},{1,2}}  {{1},{4}}    {{1,2,3}}
                                        {{2},{3}}    {{1},{5}}
                                        {{1},{1,3}}  {{2},{4}}
                                        {{2},{1,2}}  {{1},{1,4}}
                                                     {{1},{2,3}}
                                                     {{2},{1,3}}
                                                     {{3},{1,2}}
                                                     {{1},{2},{3}}
                                                     {{1},{2},{1,2}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4000
N. J. A. Sloane, Transforms

Cf. A050343-A050350, A089254.
Cf. A001970, A089259, A141268, A258466, A261049, A320328, A320330.
Row sums of A330462 and of A360764.

g:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, g[n, i-1] + If[i>n, 0, g[n-i, i-1]]]]; b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)
nn=10;Table[SeriesCoefficient[Product[(1+x^k)^PartitionsQ[k],{k,nn}],{x,0,n}],{n,0,nn}] (* Gus Wiseman, Oct 11 2018 *)

Weigh transform of A000009.

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{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}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]

A007713 Number of 4-level rooted trees with n leaves.

Original entry on oeis.org

1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Oct 11 2018: (Start)
Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((12))           ((13))
         ((1)(1))    ((111))          ((22))
         ((1))((1))  ((1)(2))         ((112))
                     ((1)(11))        ((1111))
                     ((1))((2))       ((1)(3))
                     ((1))((11))      ((2)(2))
                     ((1)(1)(1))      ((1)(12))
                     ((1))((1)(1))    ((2)(11))
                     ((1))((1))((1))  ((1)(111))
                                      ((11)(11))
                                      ((1))((3))
                                      ((2))((2))
                                      ((1))((12))
                                      ((1)(1)(2))
                                      ((2))((11))
                                      ((1))((111))
                                      ((1)(1)(11))
                                      ((11))((11))
                                      ((1))((1)(2))
                                      ((2))((1)(1))
                                      ((1))((1)(11))
                                      ((1)(1)(1)(1))
                                      ((11))((1)(1))
                                      ((1))((1))((2))
                                      ((1))((1))((11))
                                      ((1))((1)(1)(1))
                                      ((1)(1))((1)(1))
                                      ((1))((1))((1)(1))
                                      ((1))((1))((1))((1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Column k=4 of A290353.
Main diagonal of A055886.
Cf. A001970, A047968, A050342, A089259, A141268, A258466, A261049, A319066, A320328, A320330, A320331.

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: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # Alois P. Heinz, Sep 08 2008

i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Euler transform applied thrice to all-1's sequence.

A320328 Number of square multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 20, 36, 65, 117, 214, 382, 679

Offset: 0

Gus Wiseman, Oct 11 2018

A multiset partition is square if its length is equal to its number of distinct atoms.

			The a(1) = 1 through a(6) = 20 square partitions:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,1,1}}  {{2,2}}      {{1},{4}}      {{3,3}}
                  {{1},{2}}  {{1},{3}}    {{2},{3}}      {{1},{5}}
                             {{1,1,1,1}}  {{1},{1,3}}    {{2,2,2}}
                             {{1},{1,2}}  {{1},{2,2}}    {{2},{4}}
                             {{2},{1,1}}  {{2},{1,2}}    {{1},{1,4}}
                                          {{3},{1,1}}    {{4},{1,1}}
                                          {{1,1,1,1,1}}  {{1},{1,1,3}}
                                          {{1},{1,1,2}}  {{1,1},{1,3}}
                                          {{1,1},{1,2}}  {{1},{1,2,2}}
                                          {{2},{1,1,1}}  {{1,1},{2,2}}
                                                         {{1,2},{1,2}}
                                                         {{1},{2},{3}}
                                                         {{2},{1,1,2}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1},{1,1,1,2}}
                                                         {{1,1},{1,1,2}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Cf. A000219, A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A319616, A320330, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[#]==Length[Union@@#]&]],{n,8}]

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 30, 61, 110, 207, 381, 711, 1250

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 17 multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,1,1}}    {{2,2}}        {{1,1,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,2,2}}
                  {{1},{1,1}}  {{1},{3}}      {{1},{4}}
                               {{1,1,1,1}}    {{2},{3}}
                               {{1},{1,2}}    {{1,1,1,2}}
                               {{2},{1,1}}    {{1},{1,3}}
                               {{1},{1,1,1}}  {{1},{2,2}}
                                              {{2},{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}}
                                              {{1},{2},{1,1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

A320355 Number of connected antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 4, 8, 9, 19, 24, 45, 71, 118, 194, 335

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 9 clutters:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{1},{1}}  {{1,1,2}}          {{1,1,3}}
                                   {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A007718, A048143, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320330, A320351, A320353, A320356.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

A320351 Number of connected multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 38, 66, 130, 237, 449, 823, 1538

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 18 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{1,1}}    {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A319719, A320328, A320330, A320331, A320355, A320356.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[csm[#]]==1&]],{n,8}]

cp's OEIS Frontend

A050342 Expansion of Product_{m>=1} (1+x^m)^A000009(m).

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A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

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A007713 Number of 4-level rooted trees with n leaves.

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A320328 Number of square multiset partitions of integer partitions of n.

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A320331 Number of strict T_0 multiset partitions of integer partitions of n.

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A320355 Number of connected antichains of multisets whose multiset union is an integer partition of n.

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A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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A320351 Number of connected multiset partitions of integer partitions of n.

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