One-to-one correspondence between interpretations of the Tutte polynomials. (English) Zbl 1519.05130
Summary: We study relation between two interpretations of the Tutte polynomial of a matroid perspective \(M_1 \to M_2\) on a set \(E\) given with a linear ordering \(<\). A well known interpretation uses internal and external activities on a family \(\mathcal{B}(M_1, M_2)\) of the sets independent in \(M_1\) and spanning in \(M_2\). Recently we introduced another interpretation based on a family \(\mathcal{D}(M_1, M_2; <)\) of “cyclic bases” of \(M_1 \to M_2\) with respect to \(<\). We introduce a one-to-one correspondence between \(\mathcal{B}(M_1, M_2)\) and \(\mathcal{D}(M_1, M_2; <)\) that also generates a relation between the interpretations of the Tutte polynomial of a matroid perspective and corresponds with duality.
MSC:
| 05C31 | Graph polynomials |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 52B40 | Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) |
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