An Interpretation for the Tutte Polynomial

For 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 Tutteʹs coloring and flow interpretations ofTM(1 − t,0),TM(0, − t); Crapo and Rotaʹs finite field interpretation ofTM(1 − qk,0); the interpretation in terms of the Whitneycorank-nullitypolynomial; Greeneʹs interpretation as the weight enumerator of a linear code and its recent generalization to higher weight enumerators by Barg; Jaegerʹs interpretation in terms of linear code words and dual code words with disjoint support; and Brylawksi and Oxleyʹs two-variable coloring formula.