A072962 Unsigned reduced Euler characteristic for the matroid complex of cycle matroid for complete bipartite graph K_{n,n}.

1, 20, 1071, 107104, 17201225, 4053135456, 1318104508735, 565989104282624, 310299479406324369

Offset

2,2

Comments

We will denote this number by a(n,n). It is also the value of the Tutte polynomial T_{G}(0,1) for G=K_{n,n}.

The formula given for a(s,t) is valid for all s>1 and t>0. Also note that a(s,t)=a(t,s) because K_{s,t}=K_{t,s}. For small values of s we have the following formulas: a(2,t)=t-1, a(3,t)=2^{t-2}(t-1)(3t-4), a(4,t)=3^{t-3}(t-1)(16t^2-41t+27), a(5,t)=4^{t-4}(t-1)(125t^3-376t^2+378t-133)

References

W. Kook, Möbius coinvariant of complete multipartite graphs, preprint, 2002

I. Novik, A. Postnikov and B. Sturmfels: Syzygies of oriented matroids, Duke Math. J. 111 (2002), no. 2, 287-317

Links

Table of n, a(n) for n=2..10.

Formula

a(s, t)= sum_{i=0..s-2} (-1)^{i}*binomial(s-1,i)*w(s-1-i, t), where s, t>1 and an e.g.f. for w(a, b) is given by exp( sum_{i, j>0}i^{j-1}j^{i-1}(j-1)x^{i}y^{j}/i!j!).

Example

a(2,2)=1. Since K_{2,2} is a cycle with four edges, the matroid complex of cycle matroid for K_{2,2} is the 2-skeleton of standard 3-simplex. Therefore the unsigned reduced Euler characteristic for this complex is |-1+4-6+4|=1

Crossrefs

Cf. A057817.

Sequence in context: A160132 A138915 A006427 * A224125 A324416 A177596

Adjacent sequences: A072959 A072960 A072961 * A072963 A072964 A072965

Keyword

nonn

Author

W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Aug 20 2002

Status

approved