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, 211554189796535488000, 175592153482084893991151, 174356954302176729972264960, 204111110614488911169799727641, 278218647289052493421682954399744

Offset: 2

Author

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

Keywords

• nonn

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)

Examples

			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
		

References

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

Links

• Woong Kook and Kang-Ju Lee, Möbius coinvariants and bipartite edge-rooted forests, European Journal of Combinatorics, Volume 71, June 2018, Pages 180-193.
• I. Novik, A. Postnikov and B. Sturmfels, Syzygies of oriented matroids, arXiv:math/0009241 [math.CO], 2000.

Crossrefs

Cf. A057817.

Formula

a(n) = a(n, n) where 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!) ).

Extensions

More terms from Sean A. Irvine, Nov 08 2024