A320328 -id:A320328 - 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]

A096373 Number of partitions of n such that the least part occurs exactly twice.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 3, 6, 5, 11, 11, 17, 20, 30, 33, 49, 56, 77, 92, 122, 143, 190, 225, 287, 344, 435, 516, 648, 770, 951, 1134, 1388, 1646, 2007, 2376, 2868, 3395, 4078, 4807, 5749, 6764, 8042, 9449, 11187, 13101, 15463, 18070, 21236, 24772, 29021, 33764

Offset: 1

Vladeta Jovovic, Jul 19 2004

Also number of partitions of n such that the difference between the two largest distinct parts is 2 (it is assumed that 0 is a part in each partition). Example: a(6)=3 because we have [4,2], [3,1,1,1] and [2,2,2]. - Emeric Deutsch, Apr 08 2006
Number of partitions p of n+2 such that min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 27 2014
Number of partitions of n+1 such that the two smallest parts differ by one. - Giovanni Resta, Mar 07 2014
Also the number of integer partitions of n with an even number of parts that cannot be grouped into pairs of distinct parts. These are also integer partitions of n with an even number of parts whose greatest multiplicity is greater than half the number of parts. - Gus Wiseman, Oct 26 2018

			a(6)=3 because we have [4,1,1], [3,3] and [2,2,1,1].
G.f. = x^2 + 2*x^4 + x^5 + 3*x^6 + 3*x^7 + 6*x^8 + 5*x^9 + 11*x^10 + 11*x^11 + ...
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(2) = 1 through a(10) = 11 partitions where the least part occurs exactly twice (zero terms not shown):
  (11)  (22)   (311)  (33)    (322)   (44)     (522)    (55)
        (211)         (411)   (511)   (422)    (711)    (433)
                      (2211)  (3211)  (611)    (4311)   (622)
                                      (3311)   (5211)   (811)
                                      (4211)   (32211)  (3322)
                                      (22211)           (4411)
                                                        (5311)
                                                        (6211)
                                                        (33211)
                                                        (42211)
                                                        (222211)
The a(2) = 1 through a(10) = 11 partitions that cannot be grouped into pairs of distinct parts (zero terms not shown):
  (11) (22)   (2111) (33)     (2221)   (44)       (3222)     (55)
       (1111)        (3111)   (4111)   (2222)     (6111)     (3331)
                     (111111) (211111) (5111)     (321111)   (4222)
                                       (221111)   (411111)   (7111)
                                       (311111)   (21111111) (222211)
                                       (11111111)            (331111)
                                                             (421111)
                                                             (511111)
                                                             (22111111)
                                                             (31111111)
                                                             (1111111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A002865, A097091, A097092, A097093.
Cf. A117989.
Cf. A000070, A000569, A007717, A025065, A141268, A209816, A261049, A320328, A320891, A320921, A320922.

g:=sum(x^(2*k)/product(1-x^j,j=k+1..80),k=1..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 08 2006

(* do first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{p = Partitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 3]; If[ q[[1]] != q[[3]] && q[[2]] == q[[3]], c++ ]; k++ ]; c]; Table[ f[n], {n, 51}] (* Robert G. Wilson v, Jul 23 2004 *)
Table[Count[IntegerPartitions[n+2], p_ /; MemberQ[p, Length[p] + Min[p]]], {n, 50}] (* Clark Kimberling, Feb 27 2014 *)
p[n_, m_] := If[m == n, 1, If[m > n, 0, p[n, m] = Sum[p[n-m, k], {k, m, n}]]];
a[n_] := Sum[p[n+1-k, k+1], {k, n/2}]; Array[a, 100] (* Giovanni Resta, Mar 07 2014 *)

{q=sum(m=1,100,x^(2*m)/prod(i=m+1,100,1-x^i,1+O(x^60)),1+O(x^60));for(n=1,51,print1(polcoeff(q,n),","))} \\ Klaus Brockhaus, Jul 21 2004

{a(n) = if( n<0, 0, polcoeff( ( 1 - (1 - x - x^2) / eta(x + x^4 * O(x^n)) ) * (1 - x) / x^3, n))} /* Michael Somos, Feb 28 2014 */

G.f.: Sum_{m>0} (x^(2*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m)/Product_{i>m} (1-x^i)).
G.f.: Sum_{k>=1} (x^(2*k-2)*(1-x^(k-1))/Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 08 2006
a(n) = -p(n+3)+2*p(n+2)-p(n), p(n)=A000041(n). - Mircea Merca, Jul 10 2013
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Jun 02 2018

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 21 2004

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.

A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91

Offset: 1

Gus Wiseman, Nov 19 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3

			The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1

Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]

A320330 Number of T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 50, 100, 195, 366, 707, 1333, 2440

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) = 25 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,1,1}}      {{2,2}}            {{1,1,3}}
         {{1},{1}}  {{1},{2}}      {{1,1,2}}          {{1,2,2}}
                    {{1},{1,1}}    {{1},{3}}          {{1},{4}}
                    {{1},{1},{1}}  {{2},{2}}          {{2},{3}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{2},{1,1}}        {{1},{2,2}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{3},{1,1}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{2},{1,1}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, 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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@dual[#]&]],{n,8}]

A320449 Number of antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 18, 24, 39, 58, 92, 131, 206

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(7) = 24 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1},{1}}  {{1,2}}        {{1,3}}            {{1,4}}
                    {{1},{2}}      {{1},{3}}          {{2,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1},{4}}
                                   {{1},{1},{2}}      {{2},{3}}
                                   {{1},{1},{1},{1}}  {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}
.
  {{6}}                      {{7}}
  {{1,5}}                    {{1,6}}
  {{2,4}}                    {{2,5}}
  {{1,2,3}}                  {{3,4}}
  {{1},{5}}                  {{1,2,4}}
  {{2},{4}}                  {{1},{6}}
  {{3},{3}}                  {{2},{5}}
  {{1},{2,3}}                {{3},{4}}
  {{2},{1,3}}                {{1},{2,4}}
  {{3},{1,2}}                {{2},{1,4}}
  {{1},{1},{4}}              {{4},{1,2}}
  {{1,2},{1,2}}              {{1},{1},{5}}
  {{1},{2},{3}}              {{1,2},{1,3}}
  {{2},{2},{2}}              {{1},{2},{4}}
  {{1},{1},{1},{3}}          {{1},{3},{3}}
  {{1},{1},{2},{2}}          {{2},{2},{3}}
  {{1},{1},{1},{1},{2}}      {{1},{1},{2,3}}
  {{1},{1},{1},{1},{1},{1}}  {{1},{1},{1},{4}}
                             {{1},{1},{2},{3}}
                             {{1},{2},{2},{2}}
                             {{1},{1},{1},{1},{3}}
                             {{1},{1},{1},{2},{2}}
                             {{1},{1},{1},{1},{1},{2}}
                             {{1},{1},{1},{1},{1},{1},{1}}

Cf. A001970, A089259, A258466, A319719, A319721, A320328, A320353, 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]]]];
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[And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

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}]

A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{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, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, 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]]]];
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],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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A096373 Number of partitions of n such that the least part occurs exactly twice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A320330 Number of T_0 multiset partitions of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A320449 Number of antichains of sets whose multiset union is an integer partition of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords