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.

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

Author

Peter Bala, Mar 26 2012

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

			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}.
		

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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.