A035053 -id:A035053 - cp's OEIS frontend

A245566 Decimal expansion of a constant related to A007563.

4, 1, 8, 9, 6, 1, 0, 9, 5, 8, 3, 9, 3, 8, 2, 6, 9, 6, 5, 5, 2, 7, 0, 3, 6, 4, 5, 4, 5, 2, 4, 0, 4, 4, 2, 7, 5, 9, 4, 2, 3, 8, 9, 9, 2, 5, 9, 1, 5, 9, 3, 6, 5, 9, 4, 1, 3, 2, 8, 5, 7, 7, 4, 2, 5, 9, 8, 9, 8, 7, 0, 6, 4, 9, 1, 2, 0, 6, 1, 9, 9, 0, 1, 7, 6, 0, 7, 4, 0, 6, 3, 9, 5, 8, 9, 6, 8, 5, 6, 3, 3, 8, 2, 5, 3

Offset: 1

Vaclav Kotesovec, Jul 26 2014

			4.18961095839382696552703645452404427594238992591593659413285774...

Cf. A007563, A035052, A035053, A134955.

Equals lim n -> infinity A007563(n)^(1/n).
Equals lim n -> infinity A035052(n)^(1/n).
Equals lim n -> infinity A035053(n)^(1/n).
Equals lim n -> infinity A134955(n)^(1/n).

A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 48, 621, 12638, 351987, 12426060, 531225945, 26674100154, 1538781595999, 100292956964456, 7288903575373509, 584454485844541718, 51256293341752583499, 4880654469385955209092, 501471626403154217825457, 55300894427785157597436786

Offset: 0

Gus Wiseman, May 22 2018

nonn

			The a(2) = 7 hypertrees are the following:
{}
{{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

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

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) is A134958 with b(1)=1.
b(n)=if(n<2, n>=0, 2^n*sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

Binomial transform of b(1) = 1, b(n) = A134958(n) otherwise.

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 98, 263, 759, 2299, 7259, 23649, 79057, 269629, 935328, 3290260, 11714285, 42139053, 152963037, 559697097, 2062574000, 7649550572, 28534096988, 106994891146, 403119433266, 1525466082179, 5795853930652, 22102635416716, 84579153865570

Offset: 0

Gus Wiseman, May 22 2018

nonn

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

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

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) 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)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Partial sums of A035053 if we assume A035053(1) = 0.

a(n) = A304937(n) + 1 for n > 0.

A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

Original entry on oeis.org

1, 0, 1, 2, 10, 128

Offset: 0

Gus Wiseman, May 24 2018

nonn

A blob is a connected antichain of finite sets that cannot be capped by a hypertree with more than one branch.

			Non-isomorphic representatives of the a(4) = 10 blobs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

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

Cf. A030019, A035053, A048143, A054921, A134957, A144959, A275307, A286520, A293607, A304118, A304717, A304867.

A318601 Triangle read by rows: T(n,k) is the number of hypertrees on n unlabeled nodes with k edges, (0 <= k < n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 6, 7, 6, 0, 1, 3, 9, 17, 18, 11, 0, 1, 3, 13, 30, 51, 44, 23, 0, 1, 4, 17, 53, 109, 148, 117, 47, 0, 1, 4, 23, 79, 213, 372, 443, 299, 106, 0, 1, 5, 28, 119, 370, 827, 1276, 1306, 793, 235

Offset: 1

Andrew Howroyd, Aug 29 2018

Equivalently, the number of connected graphs on n unlabeled nodes with k blocks where every block is a complete graph.

Let R(x,y) be the g.f. of A318602 and S(x,y) be the g.f. of A318607. Then the number of hypertrees rooted at a vertex is R(x,y), the number rooted at an edge is y*(S(x,y) - R(x,y)) and the number rooted at a directed edge is y*S(x,y)*R(x,y). The dissymmetry theorem for trees gives that the number of unlabeled objects (this sequence) is the number rooted at a vertex plus the number rooted at an edge minus the number rooted at a directed edge.

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  2;
  0, 1, 2,  3,  3;
  0, 1, 2,  6,  7,   6;
  0, 1, 3,  9, 17,  18,  11;
  0, 1, 3, 13, 30,  51,  44,  23;
  0, 1, 4, 17, 53, 109, 148, 117,  47;
  0, 1, 4, 23, 79, 213, 372, 443, 299, 106;
  ...
Case n=4: There are 4 possible graphs which are the complete graph on 4 nodes which has 1 block, a triangle joined to a single vertex which has 2 blocks and two trees which have 3 blocks. Row 4 is then 0, 1, 1, 2.
    o---o       o---o    o---o     o--o--o
    | X |      / \       |            |
    o---o     o---o      o---o        o
.
Case n=5, k=3: The following illustrates how the dissymmetry thereom for each unlabeled hypertree gives 1 = vertex rooted + edge (block) rooted - directed edge (vertex of block) rooted.
      o---o
     / \          1 = 3 + 2 - 4
    o---o---o
.
      o   o
     / \ /        1 = 3 + 2 - 4
    o---o---o
.
      o   o
     / \ / \      1 = 4 + 3 - 6
    o---o   o
.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Rightmost diagonal is A000055 (unlabeled trees).
Row sums are A035053.
Cf. A304867, A318602, A318607.

\\ here b(n) is A318602 as vector of polynomials.
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); v}
G(n)={my(u=b(n)); apply(p->Vecrev(p), Vec(y*Ser(EulerMT(u))*(1-x*Ser(u)) + (1 - y)*Ser(u)))}
{ my(T=G(10)); for(n=1, #T, print(T[n])) }

G.f.: R(x,y) + y*(S(x,y) - R(x,y)) - y*S(x,y)*R(x,y) where R(x,y) is the g.f. of A318602 and S(x,y) is the g.f. of A318607.

A321994 Number of different chromatic symmetric functions of hypertrees on n vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 59, 165

Offset: 1

Gus Wiseman, Nov 24 2018

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).

Stanley conjectured that the number of distinct chromatic symmetric functions of trees with n vertices is equal to A000055, i.e., the chromatic symmetric function distinguishes between trees. It has been proven for trees with up to 25 vertices. If it is true in general, does the chromatic symmetric function also distinguish between hypertrees, meaning this sequence would be equal to A035053?

Jeremy L. Martin, The uniqueness problem for chromatic symmetric functions of trees (2015)
Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.

Cf. A000055, A000569, A001187, A001349, A006125, A035053, A229048, A240936, A245883, A277203, A030019, A321750, A321895, A321911.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
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]]]];
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]]]]]]]]];
density[c_]:=Total[(Length[#]-1&)/@c]-Length[Union@@c];
hyall[n_]:=Select[stableSets[Select[Subsets[Range[n]],Length[#]>1&],Or[SubsetQ[#1,#2],Length[Intersection[#1,#2]]>1]&],And[Union@@#==Range[n],Length[csm[#]]==1,density[#]==-1]&];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
Table[Length[Union[chromSF/@If[n==1,{{{1}}},hyall[n]]]],{n,5}]

A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

Original entry on oeis.org

1, 0, 1, 3, 7, 16, 38, 97, 262, 758, 2298, 7258, 23648, 79056, 269628, 935327, 3290259, 11714284, 42139052, 152963036, 559697096, 2062573999, 7649550571, 28534096987, 106994891145, 403119433265, 1525466082178, 5795853930651, 22102635416715, 84579153865569

Offset: 0

Gus Wiseman, May 21 2018

nonn

			Non-isomorphic representatives of the a(5) = 16 hypertrees are the following:
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,2,3,4,5}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{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}}

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

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304867, A304911, A304912, A304918.

\\ here b(n) 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)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)) - x)/(1-x))} \\ Andrew Howroyd, Aug 27 2018

a(n) = a(n-1) + A035053(n) for n > 1, a(n) = 1 - n for n < 2.

A304939 Number of labeled nonempty hypertrees (connected antichains with no cycles) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 0, 1, 7, 51, 506, 6843, 118581, 2504855, 62370529, 1788082153, 57997339632, 2099638691439, 83922479506503, 3670657248913385, 174387350448735877, 8942472292255441103, 492294103555090048458, 28958704109012732921523

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(3) = 7 hypertrees are the following:
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}

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

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304867, A304911, A304912, A304918, A305004.

\\ here b(n) is A030019 with b(1)=0.
b(n)=if(n<2, n==0, sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=if(n<1, n==0, sum(k=1, n, binomial(n, k)*b(k))); \\ Andrew Howroyd, Aug 27 2018

a(n) = A305004(n) - 1 for n > 0. - Andrew Howroyd, Aug 27 2018

A305004 Number of labeled hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 1, 2, 8, 52, 507, 6844, 118582, 2504856, 62370530, 1788082154, 57997339633, 2099638691440, 83922479506504, 3670657248913386, 174387350448735878, 8942472292255441104, 492294103555090048459, 28958704109012732921524

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 8 hypertrees:
{}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}

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

Cf. A030019, A035053, A134958, A134959, A304386, A304939, A304968, A304970.

\\ here b(n) is A030019 with b(1)=0.
b(n)=if(n<2, n==0, sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

a(n > 0) = A304939(n) + 1.

Binomial transform of A030019 if we assume A030019(1) = 0.

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

Offset: 1

Gus Wiseman, Jul 03 2019

A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Alois P. Heinz, Rows n = 1..185, flattened

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Edited by Peter Munn, Mar 05 2025

cp's OEIS Frontend

A245566 Decimal expansion of a constant related to A007563.

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A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

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PARI

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A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

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A305028 Number of unlabeled blobs spanning n vertices without singleton edges.

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A318601 Triangle read by rows: T(n,k) is the number of hypertrees on n unlabeled nodes with k edges, (0 <= k < n).

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A321994 Number of different chromatic symmetric functions of hypertrees on n vertices.

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Mathematica

A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

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A304939 Number of labeled nonempty hypertrees (connected antichains with no cycles) spanning some subset of {1,...,n} without singleton edges.

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A305004 Number of labeled hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.

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