An Interpretation for the Tutte Polynomial

AbstractFor any matroidMrealizable over Q, we give a combinatorial interpretation of the Tutte polynomialTM(x,y) which generalizes many of its known interpretations and specializations, including Tutte's coloring and flow interpretations ofTM(1−t,0),TM(0,−t); Crapo and Rota's finite field interpretation ofTM(1−qk,0); the interpretation in terms of the Whitneycorank-nullitypolynomial; Greene's interpretation as the weight enumerator of a linear code and its recent generalization to higher weight enumerators by Barg; Jaeger's interpretation in terms of linear code words and dual code words with disjoint support; and Brylawksi and Oxley's two-variable coloring formula