A304912 -id:A304912 - cp's OEIS frontend

A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

1, 1, 2, 8, 55, 562, 7739, 134808, 2846764, 70720278, 2021462055, 65365925308, 2359387012261, 94042995460130, 4102781803365418, 194459091322828280, 9950303194613104995, 546698973373090998382, 32101070021048906407183, 2006125858248695722280564

Offset: 0

Don Knuth, Jan 26 2008

nonn

The part of the name "assuming that each edge contains at least two vertices" is ambiguous. It may mean that not all n vertices have to be covered by some edge of the hypergraph, i.e., it is not necessarily a spanning hyperforest. However it is common to represent uncovered vertices as singleton edges, as in my example. - Gus Wiseman, May 20 2018

			From _Gus Wiseman_, May 20 2018: (Start)
The a(3) = 8 labeled spanning hyperforests are the following:
{{1,2,3}}
{{1,3},{2,3}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}
(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..370
N. J. A. Sloane, Transforms

Cf. A030019, A035053, A048143, A054921, A134955, A134956, A134957, A144959, A242817, A304716, A304717, A304867, A304911, A304912.

b:= proc(n) option remember; add(Stirling2(n-1,i) *n^(i-1), i=0..n-1) end: B:= proc(n) x-> add(b(k) *x^k/k!, k=0..n) end: a:= n-> coeff(series(exp(B(n)(x)), x, n+1), x,n) *n!: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 09 2008

b[n_] := b[n] = Sum[StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; B[n_][x_] := Sum[b[k] *x^k/k!, {k, 0, n}]; a[0]=1; a[n_] := SeriesCoefficient[ Exp[B[n][x]], {x, 0, n}] *n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Exponential transform of A030019. - N. J. A. Sloane, Jan 30 2008
Binomial transform of A304911. - Gus Wiseman, May 20 2018
a(n) = Sum of n!*Product_{k=1..n} (A030019(k)/k!)^c_k / (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) ~ exp((n+1)/LambertW(1)) * n^(n-2) / (sqrt(1+LambertW(1)) * exp(2*n+2) * (LambertW(1))^n). - Vaclav Kotesovec, Jul 26 2014

A304867 Number of non-isomorphic hypertrees of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 13, 20, 41, 70, 144, 266, 545, 1072, 2210, 4491, 9388, 19529, 41286, 87361, 186657, 399927, 862584, 1866461, 4058367, 8852686, 19384258, 42570435, 93783472, 207157172, 458805044, 1018564642, 2266475432, 5053991582, 11292781891, 25280844844

Offset: 0

Gus Wiseman, May 20 2018

nonn

A hypertree E is a connected antichain of finite sets satisfying Sum_{e in E} (|e| - 1) = |U(E)| - 1. The weight of a hypertree is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices (see A035053).
From Kevin Ryde, Feb 25 2020: (Start)
a(n), except at n=1, is the number of free trees of n edges (so n+1 vertices) where any two leaves are an even distance apart.  All trees are bipartite graphs and this condition is equivalent to all leaves being in the same bipartite half.  The diameter of a tree is always between two leaves so these trees have even diameter (A000676).
The correspondence between hypertrees and these free trees is described for instance by Bacher (start of section 1.2).  In such a free tree, call a vertex "even" if it is an even distance from a leaf.  The hypertree vertices are these even vertices.  Each hyperedge is the set of vertices surrounding an odd vertex, so hypertree weight is the total number of edges in the free tree.
(End)

			Non-isomorphic representatives of the a(6) = 5 hypertrees are the following:
  {{1,2,3,4,5,6}}
  {{1,2},{1,3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
Non-isomorphic representatives of the a(7) = 6 hypertrees are the following:
  {{1,2,3,4,5,6,7}}
  {{1,2},{1,3,4,5,6}}
  {{1,2,3},{1,4,5,6}}
  {{1,2},{1,3},{1,4,5}}
  {{1,2},{1,3},{2,4,5}}
  {{1,3},{2,4},{1,2,5}}
From _Kevin Ryde_, Feb 25 2020: (Start)
a(6) = 5 hypertrees of weight 6 and their corresponding free trees of 6 edges (7 vertices).  Each * is an "odd" vertex (odd distance to a leaf).  Each hyperedge is the set of "even" vertices surrounding an odd.
  {1,2,3,4,5,6}       3   2
                       \ /
                      4-*-1      (star 7)
                       / \
                      5   6
  .
  {1,2},{1,3,4,5}               /-3
                      2--*--1--*--4
                                \-5
  .
  {1,2,3},{1,4,5}     2-\       /-4
                         *--1--*
                      3-/       \-5
  .
  {1,2},{1,3},{1,4}    /-*--2
                      1--*--3
                       \-*--4
  .
  {1,2},{2,4},{1,3}   3--*--1--*--2--*--4   (path 7)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.

Cf. A007716, A030019, A035053, A048143, A054921, A134955, A134957, A144959, A304911, A304912, A318601.

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];
c[n_] := Module[{v = {1}}, For[i = 1, i <= Ceiling[n/2], i++, v = Join[{1}, EulerT[Join[{0}, EulerT[v]]]]]; v];
seq[n_] := Module[{u = c[n]}, x*ser[EulerT[u]]*(1 - x*ser[u]) + (1 - x)* ser[u] + x + O[x]^n // CoefficientList[#, x]&];
seq[40] (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
c(n)={my(v=[1]); for(i=1, ceil(n/2), v=concat([1], EulerT(concat([0], EulerT(v))))); v}
seq(n)={my(u=c(n)); Vec(x*Ser(EulerT(u))*(1-x*Ser(u)) + (1 - x)*Ser(u) + x + O(x*x^n))} \\ Andrew Howroyd, Aug 29 2018

a(n) = Sum_{k=1..floor(n/2)} A318601(n+1-k, k). - Andrew Howroyd, Aug 29 2018

Terms a(10) and beyond from Andrew Howroyd, Aug 29 2018

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472

Offset: 0

Gus Wiseman, May 23 2018

nonn

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

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

For n > 1, a(n) = A048143(n) * 2^n.

A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

Original entry on oeis.org

1, 2, 5, 19, 137, 3053, 822526

Offset: 0

Gus Wiseman, May 23 2018

The initial terms 1, 2, 5, 19 are the same as A304981 but the remaining terms differ.

			Non-isomorphic representatives of the a(3) = 19 clutters:
{}
{{1}}
{{1,2}}
{{1,2,3}}
{{2},{1,2}}
{{1,3},{2,3}}
{{3},{1,2,3}}
{{1},{2},{1,2}}
{{1,2},{1,3},{2,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{2},{3},{1,2,3}}
{{1},{2},{3},{1,2,3}}
{{2},{3},{1,2},{1,3}}
{{3},{1,2},{1,3},{2,3}}
{{2},{3},{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}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A304983.

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 3, 14, 118, 2916, 819473

Offset: 0

Gus Wiseman, May 23 2018

			Non-isomorphic representatives of the a(3) = 14 clutters:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{3},{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}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986, A304997.

Inverse Euler transform of A304997. - Andrew Howroyd, Aug 14 2019

a(5)-a(6) from Andrew Howroyd, Aug 14 2019

A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

Original entry on oeis.org

1, 2, 4, 12, 115, 6834, 7783198, 2414627236078, 56130437209370100252471

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 12 clutters:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

a(n > 0) = A198085(n) + 1.

a(n) = A305005(n) + n.

A304386 Number of unlabeled hypertrees (connected antichains with no cycles) spanning up to n vertices and allowing singleton edges.

Original entry on oeis.org

1, 2, 5, 15, 50, 200, 907, 4607, 25077, 144337, 863678, 5329994, 33697112, 217317986, 1424880997, 9474795661, 63769947778, 433751273356, 2977769238994, 20611559781972, 143720352656500, 1008765712435162, 7122806053951140, 50566532826530292, 360761703055959592

Offset: 0

Gus Wiseman, May 21 2018

nonn

			Non-isomorphic representatives of the a(3) = 15 hypertrees are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1},{2},{1,2}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}

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

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A134959, A144959, A304386, A304717, A304867, A304911, A304912, A304968, A304970.

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

Partial sums of b(1) = 1, b(n) = A134959(n) otherwise.

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

A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

Original entry on oeis.org

1, 1, 2, 5, 19, 176, 16118, 489996568

Offset: 0

Gus Wiseman, May 23 2018

nonn

			Non-isomorphic representatives of the a(3) = 5 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
Non-isomorphic representatives of the a(4) = 19 clutters:
  {}
  {{1,2}}
  {{1,2,3}}
  {{1,2,3,4}}
  {{1,3},{2,3}}
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Partial sums of A261006(n > 0).

A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 56, 1533, 210302, 496838435, 309068803876372, 14369391923126181310256825

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(2) = 7 clutters:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1},{2},{1,2}}

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Binomial transform of A304985(n > 0).

A304918 Number of labeled antichain hyperforests spanning a subset of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 104, 943, 12133, 203038, 4177755, 101922814, 2874725600, 92009680557, 3294276613933, 130446181101044, 5660055256165565, 267044522107706072, 13611243187516647324, 745329728016955480687, 43636132793651444511809, 2719977663069107176768790

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(3) = 18 hyperforests are the following:
{{1,2,3}}      {{2,3}}    {{1,3}}    {{1,2}}    {{3}}   {{2}}   {{1}}   {}
{{1,3},{2,3}}  {{2},{3}}  {{1},{3}}  {{1},{2}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}

Cf. A030019, A035053, A048143, A134954, A134955, A134957, A134956, A134957, A134959, A144959, A303838, A304716, A304717, A304867, A304911, A304912.

Binomial transform of A134954.

cp's OEIS Frontend

A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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A304867 Number of non-isomorphic hypertrees of weight n.

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A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304982 Number of unlabeled clutters (connected antichains) spanning up to n vertices with singleton edges allowed.

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A304983 Number of unlabeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

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A304986 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n}, if clutters of the form {{x}} are allowed for any vertex x.

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A304386 Number of unlabeled hypertrees (connected antichains with no cycles) spanning up to n vertices and allowing singleton edges.

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PARI

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A304981 Number of unlabeled clutters (connected antichains) spanning up to n vertices without singleton edges.

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A304984 Number of labeled clutters (connected antichains) spanning some subset of {1,...,n} with singleton edges allowed.

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