Dichromatic Sums Revisited

In a previous paper the author studied an enumerating power seriesΦin six variables. The typical term was the sum of a dichromatic polynomial over all rooted planar maps of given numbers of vertices and faces and given valencies for the root face and root vertex. Loops and multiple joins were allowed. The polynomial was the one now commonly called the “Tutte polynomial” by other writers. An equation forΦwas obtained. It made possible a recursive calculation of coefficients in the order of increasing edge number. The present paper arose out of the observation that the equation forΦtakes a particularly simple form when the variablesxandyof the dichromatic polynomial are each given the value 1. The value of the polynomial is then the number of spanning trees of the map concerned. In that special case a theoretical solution is obtained. It is stated in terms of a remainder obtained when a certain power series in the four remaining variables, slightly transformed, is divided by a certain polynomial.