A210586 -id:A210586 - cp's OEIS frontend

A035051 Number of labeled rooted connected graphs where every block is a complete graph.

0, 1, 2, 12, 116, 1555, 26682, 558215, 13781448, 392209380, 12641850510, 455198725025, 18109373455164, 788854833679549, 37343190699472322, 1908871649888004240, 104789417805394595600, 6148562290130009617619

Offset: 0

Christian G. Bower, Oct 15 1998

Equivalently, rooted labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).

Warren D. Smith and David Warme, Paper in preparation, 2002.

T. D. Noe, Table of n, a(n) for n=0..100
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465, 2016.
I. M. Gessel and L. H. Kalikow, Hypergraphs and a functional equation...
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 864
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees. PhD thesis, University of Virginia, 1998.

Cf. A007549, A007563, A030019, A038052, A038053, A030438. A052888, A210586.

f[n_] := Sum[ n^i*StirlingS2[n - 1, i], {i, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Apr 05 2012 *)
Table[If[n == 0, 0, BellB[n - 1, n]], {n, 0, 100}] (* Emanuele Munarini, May 23 2014 *)

a(n):=if n=0 then 0 else sum(stirling2(n-1,k)*n^k,k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, May 23 2014 */

for(n=0,30, print1(sum(k=0,n-1, stirling(n-1,k,2)*n^k), ", ")) \\ G. C. Greubel, Nov 17 2017

Recurrence: a(1) = 1, a(n) = Sum_{k=1}^{n-1} Bell(k) / k! Sum_{a_j > 0, Sum_{j=1}^k a_j = n-1} {{n-1} choose {a_1, a_2, ..., a_k }} \prod_{j=1}^k a(a_j) for n > 1, where Bell(k) = A000110(k). - Warren D. Smith, Feb 23 1998
a(n) = Sum_{i=0...n-1} S(n-1, i) n^i, where S(N, M) are Stirling numbers of the second kind - David Warme, Mar 25 1998
E.g.f. satisfies A(x)=x*exp(exp(A(x))-1).
Let X_{mu} be a Poisson random variable with mean mu: P(X_{mu} = K) = e^{-mu} mu^K / K!. The n-th moment of X_{mu} is E[X_{mu}^n] = sum_{i=0}^n S(n, i) mu^i. Therefore a(n) = E[X_n^{n-1}]. - Langworth Withers, May 25 2000
Dobinski-type formula: a(n) = 1/e^n*sum {k = 0..inf} n^k*k^(n-1)/k!. Cf. A030019 and A052888. For a refinement of this sequence see A210586. - Peter Bala, Apr 05 2012
a(n) ~ exp((1/LambertW(1)-2)*n) * n^(n-1) / (sqrt(1+LambertW(1)) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jan 22 2014

A210587 Triangle T(n,k) read by rows: T(n,k) is the number of unrooted hypertrees on n labeled vertices with k hyperedges, n >= 2, 1 <= k <= n-1.

Original entry on oeis.org

1, 1, 3, 1, 12, 16, 1, 35, 150, 125, 1, 90, 900, 2160, 1296, 1, 217, 4410, 22295, 36015, 16807, 1, 504, 19264, 179200, 573440, 688128, 262144, 1, 1143, 78246, 1240029, 6889050, 15707034, 14880348, 4782969, 1, 2550, 302500, 7770000, 69510000, 264600000, 462000000, 360000000, 100000000

Offset: 2

Peter Bala, Mar 26 2012

See A210586 for the definition of a hypertree and for the enumeration of rooted hypertrees.

			Triangle begins
.n\k.|....1.....2......3......4......5......6
= = = = = = = = = = = = = = = = = = = = = = =
..2..|....1
..3..|....1.....3
..4..|....1....12.....16
..5..|....1....35....150....125
..6..|....1....90....900...2160...1296
..7..|....1...217...4410..22295..36015..16807
...
Example of a hypertree with two hyperedges, one a 2-edge {3,4} and one a 3-edge {1,2,3}.
........__________........................
......./..........\.______................
......|....1...../.\......\...............
......|.........|.3.|....4.|..............
......|....2.....\./______/...............
.......\__________/.......................
..........................................
T(4,2) = 12. The twelve unrooted hypertrees on 4 vertices {1,2,3,4} with 2 hyperedges (one a 2-edge and one a 3-edge) have hyperedges:
{1,2,3} and {3,4}; {1,2,3} and {2,4}; {1,2,3} and {1,4};
{1,2,4} and {1,3}; {1,2,4} and {2,3}; {1,2,4} and {3,4};
{1,3,4} and {1,2}; {1,3,4} and {2,3}; {1,3,4} and {2,4};
{2,3,4} and {1,2}; {2,3,4} and {1,3}; {2,3,4} and {1,4}.

Andrew Howroyd, Table of n, a(n) for n = 2..1276
Roland Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO], 2011.
J. McCammond and J. Meier, The hypertree poset and the l^2-Betti numbers of the motion group of the trivial link, Mathematische Annalen 328 (2004), no. 4, 633-652.

Cf. A030019 (row sums). Cf. A210586, A048993.

with(combinat):
A210587 := (n, k) -> n^(k-1)*stirling2(n-1, k):
for n from 2 to 10 do seq(A210587(n, k), k = 1..n-1) end do;
# Peter Bala, Oct 28 2015

T[n_, k_] := n^(k - 1)*StirlingS2[n - 1, k];
Table[T[n, k], {n, 2, 10}, {k, 1, n - 1}] // Flatten (* Jean-François Alcover, Sep 19 2019 *)

T(n,k) = {n^(k-1)*stirling(n-1,k,2)}
for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018

T(n,k) = n^(k-1)*Stirling2(n-1,k). T(n,k) = 1/n*A210586(n,k).
E.g.f.: A(x,t) = t + x*t^2/2! + (x + 3*x^2)*t^3/3! + ..., where t*d/dt(A(x,t)) is the e.g.f. for A210586.
Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*Sum_{k = 0..inf} n^(k-1)*k^(n-1)x^k/k!.
Row sums A030019.

A367752 Number of shapes of labeled rooted hypertrees with n vertices.

Original entry on oeis.org

1, 1, 4, 29, 256, 3007, 42932, 721121, 13982563, 306967231, 7527903208, 203977383469, 6051630040496, 195111205542541, 6792697846367791, 253966747582533681, 10149075292428481965, 431705938073882999275, 19474660918369182445456, 928660364396786865580881

Offset: 1

Paul Laubie, Nov 29 2023

nonn

The shape of a labeled rooted hypertree is a labeled rooted hypertrees where we replace all the maximal subtrees by a corolla rooted on a new unlabeled black vertex.
If we remove the black vertices that are the parent of only 1 white vertex, we obtain labeled rooted hypertrees with black and white vertices such that:
  - black vertices are unlabeled;
  - black vertices have at least two children;
  - the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);
  - the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).

			For n = 3 the a(3) = 4 solutions are:
  - the corolla with a black root which have 3 white children,
  - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.

Cf. A052888, A210586, A364709, A364816.

my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ);(ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]

E.g.f.: series reversion of log(1+x)*exp(-exp(x)+x+1).

A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

Original entry on oeis.org

1, 2, 8, 55, 507, 5969, 85605, 1445420, 28110615, 618760615, 15207174501, 412790636977, 12265337498000, 395962288858946, 13800754780797740, 516494067220932259, 20658199248901273576, 879406216174705907137, 39698270229941320201019, 1894212537494300993244732

Offset: 1

Paul Laubie, Nov 29 2023

nonn

See A367752 for the definition of the shape of a labeled rooted hypertree.

Cf. A052888, A210586, A364709, A364816, A367752.

my(x='x+O('x^30)); Vec(serlaplace(serreverse((1+x)*log(1+log(1+x))*exp(-x)))) \\ Michel Marcus, Nov 30 2023

R.=PowerSeriesRing(QQ,30);(1+t)*(ln(1+ln(1+t))*exp(-t)).reverse().egf_to_ogf().list()[1:]

E.g.f.: series reversion of (1+x)*log(1+log(1+x))*exp(-x).

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A035051 Number of labeled rooted connected graphs where every block is a complete graph.

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A210587 Triangle T(n,k) read by rows: T(n,k) is the number of unrooted hypertrees on n labeled vertices with k hyperedges, n >= 2, 1 <= k <= n-1.

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A367752 Number of shapes of labeled rooted hypertrees with n vertices.

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A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

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