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We investigate the next-to-leading-colour (NLC) contributions to the colourmatrix in the fundamental and the colour-flow decompositions for tree-levelprocesses with all gluons, one quark pair and two quark pairs. By analyticalexamination of the colour factors, we find the non-zero elements in the colourmatrix at NLC. At this colour order, together with the symmetry of thephase-space, it is reduced from factorial to polynomial the scaling of thecontributing dual amplitudes as the number of partons participating in thescattering process is increased. This opens a path to an accurate tree-levelmatrix element generator of which all factorial complexity is removed, withoutresulting to Monte Carlo sampling over colour.

The colour matrix at next-to-leading-colour accuracy for tree-level multi-parton processes