A048143 -id:A048143 - cp's OEIS frontend

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

1, 2, 8, 72, 1824, 220608, 498243968, 309072306743552, 14369391925598802012151296, 146629927766168786239127150948525247729660416

Offset: 0

Gus Wiseman, May 23 2018

Only the non-singleton edges are required to form an antichain.

Number of non-degenerate unate Boolean functions of n or fewer variables. - Aniruddha Biswas, May 11 2024

			The a(2) = 8 antichains:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Aniruddha Biswas and Palash Sarkar, Counting unate and balanced monotone Boolean functions, arXiv:2304.14069 [math.CO], 2023.
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 4, 12.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A006126, A261005, A304996, A304997, A304998, A304999, A305000, A305001.
Cf. A000372, A003182, A014466, A293606, A293993, A306505, A320449, A321679.
Cf. A245079.

Binomial transform of A304999.

Inverse binomial transform of A245079. - Aniruddha Biswas, May 11 2024

a(5)-a(8) from Gus Wiseman, May 31 2018

a(9) from Aniruddha Biswas, May 11 2024

A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

Original entry on oeis.org

0, -1, -1, -2, -1, -2, -1, -3, -1, -2, -1, -3, -1, -2, -2, -4, -1, -2, -1, -3, -1, -2, -1, -4, -1, -2, -1, -3, -1, -3, -1, -5, -2, -2, -2, -3, -1, -2, -1, -4, -1, -2, -1, -3, -2, -2, -1, -5, -1, -2, -2, -3, -1, -2, -2, -4, -1, -2, -1, -4, -1, -2, -1, -6, -1, -3

Offset: 1

Gus Wiseman, May 24 2018

sign

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.
First nonnegative entry after a(1) = 0 is a(169) = 0.

			The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.
The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.
The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.
The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.

Cf. A001221, A030019, A048143, A056239, A112798, A285572, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A304911, A305001.

zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Array[zens,100]

A304118 Number of z-blobs with least common multiple n > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 19 2018

nonn

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-blob is a finite connected set S of pairwise indivisible positive integers greater than 1 such that no cap of S with at least two edges has clutter density -1.

If n is squarefree with k prime factors, then a(n) = A275307(k).

			The a(60) = 7 z-blobs together with the corresponding multiset systems (see A112798, A302242) are the following.
        (60): {{1,1,2,3}}
     (12,30): {{1,1,2},{1,2,3}}
     (20,30): {{1,1,3},{1,2,3}}
   (6,15,20): {{1,2},{2,3},{1,1,3}}
  (10,12,15): {{1,3},{1,1,2},{2,3}}
  (12,15,20): {{1,1,2},{2,3},{1,1,3}}
  (12,20,30): {{1,1,2},{1,1,3},{1,2,3}}
The a(120) = 14 z-blobs together with the corresponding multiset systems are the following.
       (120): {{1,1,1,2,3}}
     (24,30): {{1,1,1,2},{1,2,3}}
     (24,60): {{1,1,1,2},{1,1,2,3}}
     (30,40): {{1,2,3},{1,1,1,3}}
     (40,60): {{1,1,1,3},{1,1,2,3}}
   (6,15,40): {{1,2},{2,3},{1,1,1,3}}
  (10,15,24): {{1,3},{2,3},{1,1,1,2}}
  (12,15,40): {{1,1,2},{2,3},{1,1,1,3}}
  (12,30,40): {{1,1,2},{1,2,3},{1,1,1,3}}
  (15,20,24): {{2,3},{1,1,3},{1,1,1,2}}
  (15,24,40): {{2,3},{1,1,1,2},{1,1,1,3}}
  (20,24,30): {{1,1,3},{1,1,1,2},{1,2,3}}
  (24,30,40): {{1,1,1,2},{1,2,3},{1,1,1,3}}
  (24,40,60): {{1,1,1,2},{1,1,1,3},{1,1,2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A006126, A030019, A048143, A076078, A112798, A134954, A275307, A285572, A286518, A286520, A293993, A293994, A302242, A303837, A303838.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
zlobQ[s_]:=Apply[And,Composition[Not,zreeQ]/@Apply[LCM,zptns[s],{2}]];
zswell[s_]:=Union[LCM@@@Select[Subsets[s],Length[zsm[#]]==1&]];
zkernels[s_]:=Table[Select[s,Divisible[w,#]&],{w,zswell[s]}];
zptns[s_]:=Select[stableSets[zkernels[s],Length[Intersection[#1,#2]]>0&],Union@@#==s&];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[If[n==1,0,Length[Select[Rest[Subsets[Rest[Divisors[n]]]],And[zsm[#]=={n},Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={},zlobQ[#]]&]]],{n,100}]

A326750 BII-numbers of clutters (connected antichains of nonempty sets).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 292, 304, 308, 320, 512, 516, 532, 544, 548, 560, 564, 576, 768, 772, 784, 788, 800, 804, 816, 820, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304

Offset: 1

Gus Wiseman, Jul 23 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge.

			The sequence of all clutters together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}

John Tyler Rascoe, Table of n, a(n) for n = 1..6834
John Tyler Rascoe, Python program.

The number of clutters spanning n vertices is A048143(n).
Cf. A000120, A001187, A048793, A070939, A072639, A304986, A326031, A326702, A326753.
Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]

Python
```
# see linked program
```

Intersection of A326749 and A326704.

A134955 Number of "hyperforests" on n unlabeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 50, 128, 351, 1009, 3035, 9464, 30479, 100712, 340072, 1169296, 4082243, 14438577, 51643698, 186530851, 679530937, 2494433346, 9219028889, 34280914106, 128179985474, 481694091291, 1818516190252, 6894350122452

Offset: 0

Don Knuth, Jan 26 2008

nonn

A hyperforest is an antichain of finite nonempty sets (edges) whose connected components are hypertrees. It is spanning if all vertices are covered by some edge. However, it is common to represent uncovered vertices as singleton edges. For example, {{1,2},{1,4}} and {{3},{1,2},{1,4}} may represent the same hyperforest, the former being free of singletons (see example 2) and the latter being spanning (see example 1). This is different from a hyperforest with singleton edges allowed, which is one whose non-singleton edges only are required to form an antichain. For example, {{1},{2},{1,3},{2,3}} is a hyperforest with singleton edges allowed. - Gus Wiseman, May 22 2018

Equivalently, number of block graphs on n nodes, that is, graphs where every block is a complete graph. These graphs can be characterized as induced-diamond-free chordal graphs. - Falk Hüffner, Jul 25 2019

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(4) = 9 spanning hyperforests are the following:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{1,2},{3,4}}
  {{1,4},{2,3,4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{4}}
Non-isomorphic representatives of the a(4) = 9 hyperforests spanning up to 4 vertices without singleton edges are the following:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
N. J. A. Sloane, Transforms
Wikipedia, Block graph

Cf. A030019, A035053 (hypertrees), A048143, A054921, A134954 (labeled case), A134955, A134957, A144959, A245566, A304716, A304717, A304867, A304911.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(B): c:= etr(b): B:= n-> if n=0 then 0 else c(n-1) fi: C:= etr(B): aa:= proc(n) option remember; B(n)+C(n) -add(B(k)*C(n-k), k=0..n) end: a:= etr(aa): seq(a(n), n=0..27); # Alois P. Heinz, Sep 09 2008

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]; b = etr[B]; c = etr[b]; B[n_] := If[n == 0, 0, c[n-1]]; CC = etr[B]; aa[n_] := aa[n] = B[n]+CC[n]-Sum[B[k]*CC[n-k], {k, 0, n}]; a = etr[aa]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz*)

\\ here b is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); concat([1], EulerT(Vec(x*Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, May 22 2018

Euler transform of A035053. - N. J. A. Sloane, Jan 30 2008
a(n) = Sum of prod_{k=1}^n\,{A035053(k) + c_k -1 /choose c_k} over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008
a(n) ~ c * d^n / n^(5/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.36483930544... . - Vaclav Kotesovec, Jul 26 2014

A293510 Number of connected minimal covers of n vertices.

Original entry on oeis.org

1, 1, 1, 4, 23, 241, 3732, 83987, 2666729, 117807298, 7217946453, 612089089261, 71991021616582, 11761139981560581, 2675674695560997301, 849270038176762472316, 376910699272413914514283, 234289022942841270608166061, 204344856617470777364053906796

Offset: 0

Gus Wiseman, Oct 11 2017

nonn

A cover of a finite set S is a finite set of finite nonempty sets with union S. A cover is minimal if removing any edge results in a cover of strictly fewer vertices. A cover is connected if it is connected as a hypergraph or clutter. Note that minimality is with respect to covering rather than to connectedness (cf. A030019).

			The a(3) = 4 covers are: ((12)(13)), ((12)(23)), ((13)(23)), ((123)).

Cf. A030019, A046165, A048143, A275307, A283877.

nn=30;ser=Sum[(1+Sum[Binomial[n,i]*StirlingS2[i,k]*(2^k-k-1)^(n-i),{k,2,n},{i,k,n}])*x^n/n!,{n,0,nn}];
Table[n!*SeriesCoefficient[1+Log[ser],{x,0,n}],{n,0,nn}]

A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227

Offset: 1

Gus Wiseman, Nov 18 2019

nonn

A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.

			The sequence of terms tother with their corresponding clutters begins:
   1: {}              37: {{1,1,2}}            91: {{1,1},{1,2}}
   2: {{}}            41: {{6}}                97: {{3,3}}
   3: {{1}}           43: {{1,4}}             101: {{1,6}}
   5: {{2}}           47: {{2,3}}             103: {{2,2,2}}
   7: {{1,1}}         49: {{1,1},{1,1}}       107: {{1,1,4}}
   9: {{1},{1}}       53: {{1,1,1,1}}         109: {{10}}
  11: {{3}}           59: {{7}}               113: {{1,2,3}}
  13: {{1,2}}         61: {{1,2,2}}           121: {{3},{3}}
  17: {{4}}           67: {{8}}               125: {{2},{2},{2}}
  19: {{1,1,1}}       71: {{1,1,3}}           127: {{11}}
  23: {{2,2}}         73: {{2,4}}             131: {{1,1,1,1,1}}
  25: {{2},{2}}       79: {{1,5}}             137: {{2,5}}
  27: {{1},{1},{1}}   81: {{1},{1},{1},{1}}   139: {{1,7}}
  29: {{1,3}}         83: {{9}}               149: {{3,4}}
  31: {{5}}           89: {{1,1,1,2}}         151: {{1,1,2,2}}

Connected numbers are A305078.
Stable numbers are A316476.
Clutters (of sets) are A048143.
Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],And[stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]<=1]&]

Equals {1} followed by the intersection of A305078 and A316476.

A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 6, 20, 75, 310, 1422, 7094, 37877, 213610, 1256422, 7641700, 47735075, 304766742, 1981348605, 13079643892, 87480944764, 591771554768, 4042991170169, 27864757592632, 193549452132550, 1353816898675732, 9529263306483357, 67457934248821368, 480019516988969011

Offset: 0

Don Knuth, Jan 26 2008

nonn

			From _Gus Wiseman_, May 20 2018: (Start)
Non-isomorphic representatives of the a(3) = 20 hyperforests are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1},{2}}
  {{1},{2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{3},{2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A048143, A054921, A134955, A134957, A134959, A144959, A304867, A304911.

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];
EulerT[v_List] := With[{q = etr[v[[#]]&]}, q /@ Range[Length[v]]];
ser[v_] := Sum[v[[i]] x^(i - 1), {i, 1, Length[v]}] + O[x]^Length[v];
b[n_] := Module[{v = {1}}, For[i = 2, i <= n, i++, v = Join[{1}, EulerT[EulerT[2 v]]]]; v];
seq[n_] := Module[{u = 2 b[n]}, Join[{1}, EulerT[ser[EulerT[u]]*(1 - x*ser[u]) + O[x]^n // CoefficientList[#, x]&]]];
seq[24] (* Jean-François Alcover, Feb 10 2020, after Andrew Howroyd *)

\\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u)))))} \\ Andrew Howroyd, Aug 27 2018

Euler transform of A134959. - Gus Wiseman, May 20 2018

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A144959 A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 30, 78, 223, 658, 2026, 6429, 21015, 70233, 239360, 829224, 2912947, 10356334, 37205121, 134887153, 493000086, 1814902409, 6724595543, 25061885217, 93899071368, 353514105817, 1336822098961, 5075833932200

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of hyperforests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A134955(n-1) counts the hyperforests of order n with one or more isolated nodes.

			From _Gus Wiseman_, May 21 2018: (Start)
Non-isomorphic representatives of the a(5) = 11 hyperforests are the following:
  {{1,2,3,4,5}}
  {{1,2},{3,4,5}}
  {{1,5},{2,3,4,5}}
  {{1,2,5},{3,4,5}}
  {{1,2},{2,5},{3,4,5}}
  {{1,2},{3,5},{4,5}}
  {{1,4},{2,5},{3,4,5}}
  {{1,5},{2,5},{3,4,5}}
  {{1,3},{2,4},{3,5},{4,5}}
  {{1,4},{2,5},{3,5},{4,5}}
  {{1,5},{2,5},{3,5},{4,5}}
(End)

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144958 (unlabeled forests without isolated vertices), A144959, A304716, A304717, A304867, A304911.

etr[p_] := 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];
b[0] = 0; b[n_] := b[n] = etr[etr[b]][n-1];
c[1] = 0; c[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}];
a = etr[c];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 12 2018, after Alois P. Heinz's code for A035053 *)

\\ here b is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); concat([1], EulerT(concat([0], Vec(Ser(EulerT(u))*(1-x*Ser(u))-1))))} \\ Andrew Howroyd, May 22 2018

Euler transform of b(1) = 0, b(n > 1) = A035053(n). - Gus Wiseman, May 21 2018

A304887 Number of non-isomorphic blobs of weight n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 8, 14

Offset: 0

Gus Wiseman, May 20 2018

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch. The weight of a blob is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A275307).

			Non-isomorphic representatives of the a(8) = 8 blobs are the following:
  {{1,2,3,4,5,6,7,8}}
  {{1,5,6},{2,3,4,5,6}}
  {{1,2,5,6},{3,4,5,6}}
  {{1,3,4,5},{2,3,4,5}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,4},{1,5},{2,3,4,5}}
  {{2,4},{1,2,5},{3,4,5}}
  {{1,2},{1,3},{2,4},{3,4}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A035053, A048143, A275307, A283877, A286520, A293607, A293993, A293994, A304118, A304867.

cp's OEIS Frontend

A305000 Number of labeled antichains of finite sets spanning some subset of {1,...,n} with singleton edges allowed.

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A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

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Mathematica

A304118 Number of z-blobs with least common multiple n > 1.

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Mathematica

A326750 BII-numbers of clutters (connected antichains of nonempty sets).

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A134955 Number of "hyperforests" on n unlabeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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Maple

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A293510 Number of connected minimal covers of n vertices.

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A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

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A134957 Number of hyperforests with n unlabeled vertices: analog of A134955 when edges of size 1 are allowed (with no two equal edges).

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