Interpretations for the Tutte polynomials of morphisms of matroids. (English) Zbl 1498.05140
Summary: We study interpretations of the Tutte polynomials of set-pointed matroids and matroid perspectives. Suppose that \(M\) is a matroid whose ground set \(E\) is given with a linear ordering \(<\). We show that the Tutte polynomial of \(M\) pointed by \(A \subseteq E\) equals \(\sum x^{r (M / (A \cup X))} y^{r^\ast (M - (A \cup Y))} z^{r ((M / X) - Y)}\) where the sum runs through the set \(\mathcal{D} (M; A; <)\) of couples \((X, Y)\) such that \(X \subseteq E \setminus A\), \(Y = (E \setminus A) \setminus X\), and \(X \cap C\) (resp. \(Y \cap C)\) differs from \(\{\min (C) \}\) for each circuit \(C\) of \(M^\ast - A\) (resp. \(M - A)\). Furthermore, we characterize \(\mathcal{D} (M; A; <)\) by a contraction-deletion rule. Analogous results are introduced for the Tutte polynomials of matroid perspectives.
Keywords:
Tutte polynomial; set-pointed matroid; matroid perspective; \((M,<)\)-compatible set; dualityReferences:
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