A055779 Number of fat trees on n labeled vertices.

1, 2, 10, 89, 1156, 19897, 428002, 11067457, 334667368, 11593751921, 452892057454, 19699549177585, 944416040000044, 49480473036710185, 2812998429218735986, 172475808692526176513, 11345688093224067380176

Offset: 1

Author

Thomas Zaslavsky, Jul 12 2000

Keywords

• nonn
• nice
• easy

Comments

A fat tree on vertex set V is a partition of V together with edges (between vertices, not parts) that link the parts of the partition in a tree-like pattern: that is, when the parts are collapsed to points, the edges are a (free) tree. A fat tree is in a (multi)graph G when the edges are edges of G. The fat forests in a graph form a geometric lattice.

If a(n) is the number of fat trees when each edge is replaced by M distinguishable copies of itself, then a(1) = 1, a(2) = M + 1, a(3) = 3 M^2 + 6 M + 1, a(4) = 16 M^3 + 48 M^2 + 24 M + 1, a(5) = 125 M^4 + 500 M^3 + 450 M^2 + 80 M + 1, a(6) = 1296 M^5 + 6480 M^4 + 8640 M^3 + 3240 M^2 + 240 M + 1.

Examples

			For n=3, there is one fat tree with a single node, three with three nodes (choose which vertex to have in the middle) and six with two nodes (3 choices for which vertex to have by itself and 2 choices for which of the others to join it to).
		

References

• Thomas Zaslavsky, "Perpendicular dissections of space". Discrete Comput. Geom., 27 (2002), 303-351. MR 2003i:52026. Zbl. 1001.52011.

Links

• T. D. Noe, Table of n, a(n) for n=1..100
• Vaclav Kotesovec, Asymptotic formula for number of fat trees on n labeled vertices, Aug 25 2012, in Czech, main results in English.
• T. Zaslavsky, Perpendicular dissections of space, arXiv:1001.4435 [math.CO], 2010.

Programs

Mathematica

Table[Sum[Binomial[n,k]k^(n-k) n^(k-2),{k,n}],{n,20}] (* Harvey P. Dale, Aug 24 2016 *)

PARI

A055779(n) = sum(k=1, n, binomial(n, k)*k^(n-k)*n^(k-2)) \\ Franklin T. Adams-Watters, Jun 16 2006

Formula

a(n) = Sum_{k=1..n} binomial(n, k)*k^(n-k)*n^(k-2). - Vladeta Jovovic, Jun 16 2006

a(n) = n!/n^2 sum_{mu a partition of n} product_j n^{mu_j}/(mu_j! (j-1)!^{mu_j}), where mu_j is the number of parts of size j in the partition mu. - Vladeta Jovovic, Jun 15 2006

Lim_{n->infinity} (a(n)^(1/n))/n = (1-p)^(p-1)*p^(1-2*p) ~ 1.6554879129915343..., where p ~ 0.6924583254616546... is the root of the equation exp(1-1/p)=(1-p)/p^2. - Vaclav Kotesovec, Aug 25 2012

Extensions

Edited with more terms by Franklin T. Adams-Watters, Jun 13 2006

More terms from Vladeta Jovovic and Franklin T. Adams-Watters, Jun 15 2006