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Electronic transport through chaotic quantum dots exhibits universalbehaviour which can be understood through the semiclassical approximation.Within the approximation, transport moments reduce to codifying classicalcorrelations between scattering trajectories. These can be represented asribbon graphs and we develop an algorithmic combinatorial method to generateall such graphs with a given genus. This provides an expansion of the lineartransport moments for systems both with and without time reversal symmetry. Thecomputational implementation is then able to progress several orders higherthan previous semiclassical formulae as well as those derived from anasymptotic expansion of random matrix results. The patterns observed alsosuggest a general form for the higher orders.

Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions