A083483 Number of forests with two connected components in the complete graph K_{n}.

0, 1, 3, 15, 110, 1080, 13377, 200704, 3542940, 72000000, 1656409535, 42568187904, 1208912928522, 37603105146880, 1271514111328125, 46443371157258240, 1822442358054692408, 76461926986744528896, 3415753581721829617275

Offset: 1

Author

Woong Kook (andrewk(AT)math.uri.edu), Jun 08 2003

Keywords

• nonn

Comments

Note that the above sequence is dominated by the sequence n^{n-2} (n > 0), A000272, which enumerates the number of spanning trees in K_{n} : 1, 1, 3, 16, 125, 1296, 16807, 262144, ... This is a consequence of the result in [EKT] which shows that the sequence of independent set numbers of cycle matroid of K_{n} is (strictly) monotone increasing (when n > 3).

References

• W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• N. Eaton, W. Kook, and L. Thoma, Monotonicity for complete graphs, preprint
• A. Kassel, R. Kenyon, and W. Wu, Random two-component spanning forests, Ann. Inst. H. Poincaré Probab. Statist., 51 (2015), 1457-1464.
• C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algebraic Discrete Methods, 5 (1984), no. 3, 384--406. MR0752043 (86d:05059). See Eq. (47). - From _N. J. A. Sloane_, Apr 09 2014

Crossrefs

Cf. A000272, A053506, A239910.

Column m=2 of A105599. A diagonal of A138464. - Alois P. Heinz, Apr 10 2014

Programs

Magma

[n^(n-4)*(n-1)*(n+6)/2 : n in [1..20]]; // Vincenzo Librandi, Apr 10 2014

Maple

f:=n->(n-1)!*n^(n-4)*(n+6)/(2*(n-2)!); [seq(f(n),n=2..30)]; # N. J. A. Sloane, Apr 09 2014

Mathematica

(* first 20 terms starting with n=1 *) T := Sum[i^(i - 2)*(x^i)/i!, {i, 1, 20}]; T2 := Expand[(T^{2})/2! ]; C2[i_] := Coefficient[T2, x^{i}]*i!; M := MatrixForm[Table[C2[i], {i, 20}]]; M
Table[n^(n - 4) (n - 1) (n + 6)/2, {n, 1, 40}] (* Vincenzo Librandi, Apr 10 2014 *)

PARI

for(n=1,30, print1(n^(n-4)*(n-1)*(n+6)/2, ", ")) \\ G. C. Greubel, Nov 14 2017

Formula

E.g.f.: T(x)^{2}/2!, where T(x) is the e.g.f. for the number of spanning trees in K_{n}, i.e., T(x) = Sum_{i>=1} i^(i-2)*x^i/i!.

E.g.f.: (1/8)*LambertW(-x)^2*(2+LambertW(-x))^2. - Vladeta Jovovic, Jul 08 2003

a(n) = n^(n-4)*(n-1)*(n+6)/2. - Vaclav Kotesovec, Oct 18 2013

Extensions

Edited by N. J. A. Sloane, Apr 09 2014