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Tutte polynomials and bicycle dimension of ternary matroids. (English) Zbl 0669.05023
Let $$M$$ be a ternary matroid, $$t(M,x,y)$$ be its Tutte polynomial and $$d(M)$$ be the dimension of the bicycle space of any representation of $$M$$ over $$GF(3)$$.
We show that, for $$j=e^{2i\pi /3}$$, the modulus of the complex number $$t(M,j,j^ 2)$$ is equal to ($$\sqrt{3})^{d(M)}$$. The proof relies on the study of the weight enumerator $$W_{{\mathcal C}}(y)$$ of the cycle space $${\mathcal C}$$ of a representation of $$M$$ over $$GF(3)$$ evaluated at $$y=j$$. The main tool is the concept of principal quadripartition of $${\mathcal C}$$ which allows a precise analysis of the evolution of the relevant invariants under deletion and contraction of elements.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices 05A15 Exact enumeration problems, generating functions
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