

A072962


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


0




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)=t1, a(3,t)=2^{t2}(t1)(3t4), a(4,t)=3^{t3}(t1)(16t^241t+27), a(5,t)=4^{t4}(t1)(125t^3376t^2+378t133)


REFERENCES

W. Kook, MÃ¶bius 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, 287317


LINKS

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


FORMULA

a(s, t)= sum_{i=0..s2} (1)^{i}*binomial(s1,i)*w(s1i, t), where s, t>1 and an e.g.f. for w(a, b) is given by exp( sum_{i, j>0}i^{j1}j^{i1}(j1)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 2skeleton of standard 3simplex. Therefore the unsigned reduced Euler characteristic for this complex is 1+46+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



