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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

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

LINKS

Table of n, a(n) for n=2..43.

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-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.

EXAMPLE

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).

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

CROSSREFS

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

Sequence in context: A291418 A219512 A186695 * A019232 A185697 A263008

Adjacent sequences:  A210584 A210585 A210586 * A210588 A210589 A210590

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Mar 26 2012

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.