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A072962
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Unsigned reduced Euler characteristic for the matroid complex of cycle matroid for complete bipartite graph K_{n,n}.
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0
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OFFSET
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2,2
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COMMENTS
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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)
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REFERENCES
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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
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LINKS
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FORMULA
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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!).
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Aug 20 2002
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STATUS
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approved
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