A317795 -id:A317795 - cp's OEIS frontend

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.
Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}

			The a(4) = 12 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}

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

The set-system version is A330054 (no endpoints) or A306005 (no singletons).
Non-isomorphic multiset partitions are A007716.
Set-systems with no singletons are A016031.
Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

\\ compare with similar program for A007716.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023

Extended by Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

A317794 Number of non-isomorphic set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 1, 2, 8, 180, 612032, 200253854316544, 263735716028826427534807159537664, 5609038300883759793482640992086670066760184863720423808367168537493504

Offset: 0

Gus Wiseman, Aug 07 2018

nonn

			Non-isomorphic representatives of the a(3) = 8 set-systems:
  0,
  {12}, {123},
  {12}{13}, {12}{123},
  {12}{13}{23}, {12}{13}{123},
  {12}{13}{23}{123}.

Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.
Peter H. van der Kamp, Hypergraphs and homogeneous Lotka-Volterra systems with linear Darboux polynomials, arXiv:2411.18264 [nlin.SI], 2024. See p. 4.

The spanning case is A317795.

Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876, A000616, A000370.

sysnorm[{}] := {};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]>1&]],Or[Length[#]==0,Union@@#==Range[Max@@Union@@#]]&]]],{n,4}]
(* second program *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{2,n}]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length]/n!,{prm,Permutations[Range[n]]}],{n,6}] (* Gus Wiseman, Dec 12 2018 *)

a(n) = A000616(n) - A000370(n). - Tilman Piesk, Apr 14 2025

More terms from Gus Wiseman, Dec 12 2018

A323819 Number of non-isomorphic connected set-systems covering n vertices.

Original entry on oeis.org

1, 1, 3, 30, 1912, 18662590, 12813206131799685, 33758171486592987138461432668177794, 1435913805026242504952006868879460423767388571975632398910903473535427583

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			Non-isomorphic representatives of the a(3) = 30 set-systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{3},{2,3},{1,2,3}}
  {{2},{3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Alois P. Heinz, Table of n, a(n) for n = 0..12
Thomas Delacroix, Meaningful objective frequency-based interesting pattern mining, Thesis, 2021.
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.

Cf. A000295, A003465, A016031, A048143, A055621 (not necessarily connected), A293510, A317795, A323817, A323818 (labeled case).

nmax = 12;
b[n_, i_, l_] := b[n, i, l] = If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
f[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]]/2;
A055621 = f /@ Range[0, nmax];
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
Join[{1}, EULERi[A055621 // Rest]] (* Jean-François Alcover, Jan 31 2020, after Alois P. Heinz in A055621 *)

Inverse Euler transform of A055621.

A320395 Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.

Original entry on oeis.org

1, 2, 10, 208, 45960, 287800704, 100103176111616, 3837878984050795692032, 32966965900633495618246298767360, 128880214965936601447070466061615999984402432, 464339910355487357558396669850788946402420533504952464572416

Offset: 0

Gus Wiseman, Dec 12 2018

nonn

			Non-isomorphic representatives of the a(2) = 10 multiset systems:
  {}
  {{111}}
  {{122}}
  {{111}{222}}
  {{112}{122}}
  {{112}{222}}
  {{122}{222}}
  {{111}{122}{222}}
  {{112}{122}{222}}
  {{111}{112}{122}{222}}

Andrew Howroyd, Table of n, a(n) for n = 0..25

The 2-uniform case is A000666. The case of sets (as opposed to multisets) is A000665. The case of labeled spanning sets is A302374, with unlabeled case A322451.

Cf. A000088, A000612, A003180, A070166, A301922, A317795, A319876.

Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Select[Tuples[Range[n],3],OrderedQ]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,6}]

permcount(v)={my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(perm)={my(t=0); forsubset([#perm+2, 3], v, t += can([v[1],v[2]-1,v[3]-2], t->perm[t])); t}
a(n)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(rep(p))); s/n!} \\ Andrew Howroyd, Aug 26 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 26 2019

A323817 Number of connected set-systems covering n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 1990, 67098648, 144115187673201808, 1329227995784915871895000743748659792, 226156424291633194186662080095093570015284114833799899656335137245499581360

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			The a(3) = 12 set-systems:
  {{1, 2, 3}}
  {{1, 2}, {1, 3}}
  {{1, 2}, {2, 3}}
  {{1, 3}, {2, 3}}
  {{1, 2}, {1, 2, 3}}
  {{1, 3}, {1, 2, 3}}
  {{2, 3}, {1, 2, 3}}
  {{1, 2}, {1, 3}, {2, 3}}
  {{1, 2}, {1, 3}, {1, 2, 3}}
  {{1, 2}, {2, 3}, {1, 2, 3}}
  {{1, 3}, {2, 3}, {1, 2, 3}}
  {{1, 2}, {1, 3}, {2, 3},{1, 2, 3}}
The A323816(4) - a(4) = 3 disconnected set-systems covering n vertices with no singletons:
  {{1, 2}, {3, 4}}
  {{1, 3}, {2, 4}}
  {{1, 4}, {2, 3}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A001187, A016031, A048143, A092918, A293510, A317795, A323816 (not necessarily connected), A323818 (with singletons), A323819, A323820 (unlabeled case).

m:=10;
A323816:= func< n | (&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]) >;
f:= func< x | 1 + Log((&+[A323816(j)*x^j/Factorial(j): j in [0..m+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 05 2022

b:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

nn=10;
ser=Sum[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}]*x^n/n!,{n,0,nn}];
Table[SeriesCoefficient[1+Log[ser],{x,0,n}]*n!,{n,0,nn}]

m=10
def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
def A323817_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1 + log(sum(A323816(j)*x^j/factorial(j) for j in range(m+2))) ).egf_to_ogf().list()
A323817_list(m) # G. C. Greubel, Oct 05 2022

Logarithmic transform of A323816.

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A323816 Number of set-systems covering n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 1993, 67098768, 144115187673233113, 1329227995784915871895000745158568460, 226156424291633194186662080095093570015284114833799899660370362545578585265

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			The a(3) = 12 set-systems:
  {{1,2,3}}
  {{1,2}, {1,3}}
  {{1,2}, {2,3}}
  {{1,3}, {2,3}}
  {{1,2}, {1,2,3}}
  {{1,3}, {1,2,3}}
  {{2,3}, {1,2,3}}
  {{1,2}, {1,3}, {2,3}}
  {{1,2}, {1,3}, {1,2,3}}
  {{1,2}, {2,3}, {1,2,3}}
  {{1,3}, {2,3}, {1,2,3}}
  {{1,2}, {1,3}, {2,3}, {1,2,3}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A000295, A000371, A000612, A003465 (with singletons), A006129 (covers by pairs), A016031, A055154, A055621, A305001, A317795 (unlabeled case), A323817 (connected case).

[(&+[(-1)^(n-j)*Binomial(n,j)*2^(2^j -j-1): j in [0..n]]): n in [0..12]]; // G. C. Greubel, Oct 05 2022

a:= n-> add(2^(2^(n-j)-n+j-1)*binomial(n, j)*(-1)^j, j=0..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

Table[Sum[(-1)^(n-k)*Binomial[n,k]*2^(2^k-k-1),{k,0,n}],{n,0,8}]

def A323816(n): return sum((-1)^j*binomial(n,j)*2^(2^(n-j) -n+j-1) for j in range(n+1))
[A323816(n) for n in range(12)] # G. C. Greubel, Oct 05 2022

Inverse binomial transform of A016031 shifted once to the left.

A319876 Irregular triangle read by rows where T(n,k) is the number of permutations of {1,...,n} whose action on 2-element subsets of {1,...,n} has k cycles.

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 1, 0, 0, 14, 0, 9, 0, 1, 0, 0, 24, 50, 20, 0, 15, 10, 0, 0, 1, 0, 0, 0, 264, 0, 340, 0, 40, 0, 60, 0, 15, 0, 0, 0, 1, 0, 0, 0, 720, 1764, 504, 0, 1120, 630, 0, 0, 70, 105, 105, 0, 0, 21, 0, 0, 0, 0, 1, 0, 0, 0, 0, 13488, 0, 14112, 0, 3724, 0

Offset: 1

Gus Wiseman, Dec 12 2018

The permutation
  1 -> 1
  2 -> 2
  3 -> 4
  4 -> 3
acts on unordered pairs of distinct elements of {1,2,3,4} to give
  (1,2) -> (1,2)
  (1,3) -> (1,4)
  (1,4) -> (1,3)
  (2,3) -> (2,4)
  (2,4) -> (2,3)
  (3,4) -> (3,4)
which has 4 cycles
       (1,2)
  (1,3) <-> (1,4)
  (2,3) <-> (2,4)
       (3,4)
so is counted under T(4,4).

			Triangle begins:
   1
   0   2
   0   2   3   1
   0   0  14   0   9   0   1
   0   0  24  50  20   0  15  10   0   0   1
   0   0   0 264   0 340   0  40   0  60   0  15   0   0   0   1
The T(4,4) = 9 permutations: (1243), (1324), (1432), (2134), (2143), (3214), (3412), (4231), (4321).

Row n has A000124(n - 1) terms. Row sums are the factorial numbers A000142.

Cf. A000088, A000612, A000665, A000666, A003180, A050535, A070166, A317794, A317795.

Table[Length[Select[Permutations[Range[n]],PermutationCycles[Ordering[Map[Sort,Subsets[Range[n],{2}]/.Rule@@@Table[{i,#[[i]]},{i,n}],{1}]],Length]==k&]],{n,5},{k,0,n*(n-1)/2}]

A000088(n) = (1/n!) * Sum_k 2^k * T(n,k).

A330124 Number of unlabeled set-systems with n vertices and no endpoints.

Original entry on oeis.org

1, 1, 2, 22, 1776

Offset: 0

Gus Wiseman, Dec 05 2019

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

			Non-isomorphic representatives of the a(3) = 22 set-systems:
  0
  {1}{2}{12}
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Partial sums of the covering case A330196.
The labeled version is A330059.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems with no singletons are A317794.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
The case with no singletons is A320665.
Cf. A007716, A016031, A306005, A317795, A321405, A330052, A330055.

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1

Offset: 0

Gus Wiseman, Dec 28 2023

A set-system is a finite set of finite nonempty sets.

Conjecture: Column k = 2 is A101881.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

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

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

cp's OEIS Frontend

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

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A317794 Number of non-isomorphic set-systems on n vertices with no singletons.

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A323819 Number of non-isomorphic connected set-systems covering n vertices.

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A320395 Number of non-isomorphic 3-uniform multiset systems over {1,...,n}.

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A323817 Number of connected set-systems covering n vertices with no singletons.

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A368186 Number of n-covers of an unlabeled n-set.

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A323816 Number of set-systems covering n vertices with no singletons.

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