OFFSET
2,1
COMMENTS
A hypergraph H is a pair (V,E) consisting of a finite set V of vertices and a set E of hyperedges given by subsets of V containing at least two elements. A walk in a hypergraph H connecting vertices v0 and vn is a sequence v0, e1, v1, e2, ... , v(n-1), en, vn, where each vi is in V and each ei is in E and for each ei the set {v(i-1),vi} is contained in ei. If for every pair of vertices v and v0 there is a walk in H starting at v and ending at v0 then H is called connected. A walk is a cycle if it contains at least two edges, all of the ei are distinct and all of the vi are distinct except v0 = vn. A connected hypergraph with no cycles is called a hypertree. A rooted hypertree is a hypertree in which one particular vertex is selected as being the root. For the enumeration of unrooted hypertrees see A210587.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..1276
R. 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.
FORMULA
T(n,k) = n^k*Stirling2(n-1,k). T(n,k) = n*A210587(n,k).
E.g.f. A(x,t) = t + 2*x*t^2/2! + (3*x + 9*x^2)*t^3/3! + ... satisfies A(x,t) = t*exp(x*(exp(A(x,t)) - 1)).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*Sum_{k = 0..inf} n^k*k^(n-1)x^k/k!.
Row sums A035051.
The e.g.f. is essentially the series reversion of t/F(x,t) w.r.t. t, where F(x,t) = exp(x*(exp(t) - 1)) is the e.g.f. of the Stirling numbers of the second kind A048993. - Peter Bala, Oct 28 2015
EXAMPLE
Triangle begins
.n\k.|....1.....2......3.......4.......5.......6
= = = = = = = = = = = = = = = = = = = = = = = = =
..2..|....2
..3..|....3.....9
..4..|....4....48.....64
..5..|....5...175....750.....625
..6..|....6...540...5400...12960....7776
..7..|....7..1519..30870..156065..252105..117649
...
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) = 48. 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}.
Choosing one of the four vertices as the root gives a total of 4x12 = 48 rooted hypertrees on 4 vertices.
MAPLE
with(combinat):
A210586 := (n, k) -> n^k*stirling2(n-1, k):
for n from 2 to 10 do seq(A210586(n, k), k = 1..n-1) end do;
# Peter Bala, Oct 28 2015
PROG
(PARI) T(n, k) = {n^k*stirling(n-1, k, 2)}
for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Mar 26 2012
STATUS
approved