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A055779
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Number of fat trees on n labeled vertices.
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2
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1, 2, 10, 89, 1156, 19897, 428002, 11067457, 334667368, 11593751921, 452892057454, 19699549177585, 944416040000044, 49480473036710185, 2812998429218735986, 172475808692526176513, 11345688093224067380176
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| Thomas Zaslavsky, "Perpendicular dissections of space". Discrete Comput. Geom., 27 (2002), 303-351. MR 2003i:52026. Zbl. 1001.52011.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
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FORMULA
| 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
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EXAMPLE
| 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).
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PROG
| (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
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CROSSREFS
| Sequence in context: A060350 A096658 A186184 * A198434 A179423 A067550
Adjacent sequences: A055776 A055777 A055778 * A055780 A055781 A055782
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jul 12 2000
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EXTENSIONS
| 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
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