

A210586


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


3



2, 3, 9, 4, 48, 64, 5, 175, 750, 625, 6, 540, 5400, 12960, 7776, 7, 1519, 30870, 156065, 252105, 117649, 8, 4032, 154112, 1433600, 4587520, 5505024, 2097152, 9, 10287, 704214, 11160261, 62001450, 141363306, 133923132, 43046721, 10, 25500, 3025000, 77700000, 695100000, 2646000000, 4620000000, 3600000000, 1000000000
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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(n1), en, vn, where each vi is in V and each ei is in E and for each ei the set {v(i1),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^2Betti numbers of the motion group of the trivial link, Mathematische Annalen 328 (2004), no. 4, 633652.


FORMULA

T(n,k) = n^k*Stirling2(n1,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)).
Dobinskitype formula for the row polynomials: R(n,x) = exp(n*x)*sum {k = 0..inf} n^k*k^(n1)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 2edge {3,4) and one a 3edge{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 2edge and one a 3edge) 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(n1, k):
for n from 2 to 10 do seq(A210586(n, k), k = 1..n1) end do;
# Peter Bala, Oct 28 2015


PROG

(PARI) T(n, k) = {n^k*stirling(n1, k, 2)}
for(n=2, 10, for(k=1, n1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018


CROSSREFS

Cf. A035051 (row sums). Cf. A210587, A048993.
Sequence in context: A249824 A227912 A229212 * A202017 A127198 A065631
Adjacent sequences: A210583 A210584 A210585 * A210587 A210588 A210589


KEYWORD

nonn,easy,tabl


AUTHOR

Peter Bala, Mar 26 2012


STATUS

approved



