A100070 Number a(n) of forests with two components in the complete bipartite graph K_{n,n}.

6, 117, 5632, 515625, 77262336, 17230990189, 5360119185408, 2219048868131217, 1180000000000000000, 783948341202404638821, 636404158746280870281216, 619884903445287035295372217, 713552333492738487958741450752

Offset: 2

Author

Woong Kook (andrewk(AT)math.uri.edu), Nov 02 2004

Keywords

• nonn

Comments

This sequence (a(n)) appears to dominate the sequence (n^{2n-2}) of the number of spanning trees in K_{n,n} for n>1. This shows that the sequence of independent set numbers for the cycle matroid of K_{n,n} is not monotone increasing unlike the complete graph K_{n}.

Examples

			a(2)=6 because K_{2,2} is C_{4} the cycle of length 4 and there are 6 forests with two components in C_{4}.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 2..214
• N. Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs, preprint, 2004.

Crossrefs

Cf. A000272, A069087, A083483.

Programs

Mathematica

a[n_]:=Sum[Binomial[n, x]*Binomial[n, y]*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1), {x, 1, n-1}, {y, 1, n-1}]/2 + (2*(n^2-n)^(n-1)); Table[a[n], {n, 2, 10}] (* This will generate a(n) from n=2 to 10. *)

Formula

a(n) = 2*(n^2 - n)^(n-1) + (1/2)*Sum_{x=1..(n-1)} Sum_{y=1..(n-1)} b(n, x, y), where b(n, x, y) = binomial(n,x)*binomial(n,y)*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1).