1,2

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.

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

T. D. Noe, Table of n, a(n) for n=1..100

a(n) = Sum_{k=1..n} binomial(n, k)*k^(n-k)*n^(k-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), Jun 15 2006

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).

(PARI) A055779(n) = local(k); sum(k=1, n, binomial(n, k)*k^(n-k)*n^(k-2)). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 16 2006

Sequence in context: A144002 A060350 A096658 this_sequence A067550 A086587 A082472

Adjacent sequences: A055776 A055777 A055778 this_sequence A055780 A055781 A055782

nonn,nice,easy

Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jul 12 2000

Edited with more terms by Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 13 2006

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 15 2006