

A030019


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


80



1, 1, 1, 4, 29, 311, 4447, 79745, 1722681, 43578820, 1264185051, 41381702275, 1509114454597, 60681141052273, 2667370764248023, 127258109992533616, 6549338612837162225, 361680134713529977507, 21333858798449021030515, 1338681172839439064846881
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OFFSET

0,4


COMMENTS

Equivalently, this is the number of "hypertrees" on n labeled nodes, i.e. connected hypergraphs that have no cycles, assuming that each edge contains at least two vertices.  Don Knuth, Jan 26 2008. See A134954 for hyperforests.
Also number of labeled connected graphs where every block is a complete graph (cf. A035053).
Let H = (V,E) be the complete hypergraph on N labeled vertices (all edges having cardinality 2 or greater). Let e in E and K = e. Then the number of distinct spanning trees of H that contain edge e is g(N,K) = K * E[X_N^{NK}] / N and the K=1 case gives this sequence. Clearly there is some deep structural connection between spanning trees in hypergraphs and Poisson moments.


REFERENCES

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


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..370 (first 101 terms from T. D. Noe)
Ayomikun Adeniran and Catherine Yan, Gončarov Polynomials in Partition Lattices and Exponential Families, arXiv:1907.07814 [math.CO], 2019.
Ronald Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708v1 [math.CO], 2011.
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the SplitDecomposition, arXiv:1608.01465 [math.CO], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 810.
Louis H. Kalikow, Enumeration of parking functions, allowable permutation pairs, and labeled trees, PhD thesis, Brandeis University, 1999.
R. Lorentz, S. Tringali, and C.H. Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039, 2015.
Adam Piggott, The symmetries of McculloughMiller space, 2011, preprint.
Adam Piggott, The symmetries of McculloughMiller space, Algebra and Discrete Mathematics 14(2) (2012), 239266.
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees, PhD thesis, University of Virginia, 1998, Table 5.1.
D. M. Warme, Spanning Trees in Hypergraphs with Applications to Steiner Trees, PhD thesis, University of Virginia, 1998, Table 5.1.
Index entries for sequences related to trees


FORMULA

a(n) = A035051(n)/n for n > 0.
a(n) = Sum_{i=0...n1} Stirling2(n1, i) n^(i1), n >= 1. (Warme, Corollary 3.15.1, p. 59)
a(n) = E[X_n^{n1}] / n, n >= 1, where X_n is a Poisson random variable with mean n.
1 = Sum_{n>=0} a(n+1) * x^n/n! * exp( (n+1)*(exp(x)1) ).  Paul D. Hanna, Jun 11 2011
E.g.f. satisfies: A(x) = Sum_{n>=0} exp(n*x*A(x)1)/n! = Sum_{n>=0} a(n+1)*x^n/n!.  Paul D. Hanna, Sep 25 2011
Dobinskitype formula: a(n) = 1/e^n*sum {k = 0..inf} n^(k1)*k^(n1)/k!. Cf. A052888. For a refinement of this sequence see A210587.  Peter Bala, Apr 05 2012
a(n) ~ n^(n2) / (sqrt(1+LambertW(1)) * (LambertW(1))^(n1) * exp((21/LambertW(1))*n)).  Vaclav Kotesovec, Jul 26 2014


MATHEMATICA

a[n_] := Sum[ StirlingS2[n1, i]*n^(i1), {i, 0, n1}]; a[0] = 1; Table[a[n], {n, 0, 18}](* JeanFrançois Alcover, Sep 12 2012, from 2nd formula *)


PROG

(PARI) {a(n)=if(n==0, 1, (n1)!*polcoeff(1sum(k=0, n2, a(k+1)*x^k/k!*exp((k+1)*(exp(x+O(x^n))1))), n1))} /* Paul D. Hanna */
(PARI) /* E.g.f. of sequence shifted left one place: */
{a(n)=local(A=1+x); for(i=1, n, A=exp(1)*sum(m=0, 2*n+10, exp(m*x*A+x*O(x^n))/m!)); round(n!*polcoeff(A, n))} /* Paul D. Hanna */


CROSSREFS

Cf. A030438, A035051, A035053, A134954, A134956, A134958. A052888, A210587.
Sequence in context: A014622 A067146 A210949 * A303928 A201627 A195194
Adjacent sequences: A030016 A030017 A030018 * A030020 A030021 A030022


KEYWORD

nonn,nice


AUTHOR

David Warme (warme(AT)s3i.com)


EXTENSIONS

More terms, formula and comment from Christian G. Bower Dec 15 1999


STATUS

approved



