  The non-Abelian exponentiation theorem has recently been generalised to
correlators of multiple Wilson line operators. The perturbative expansions of
these correlators exponentiate in terms of sets of diagrams called webs, which
together give rise to colour factors corresponding to connected graphs. The
colour and kinematic degrees of freedom of individual diagrams in a web are
entangled by mixing matrices of purely combinatorial origin. In this paper we
relate the combinatorial study of these matrices to properties of partially
ordered sets (posets), and hence obtain explicit solutions for certain families
of web-mixing matrix, at arbitrary order in perturbation theory. We also
provide a general expression for the rank of a general class of mixing
matrices, which governs the number of independent colour factors arising from
such webs. Finally, we use the poset language to examine a previously
conjectured sum rule for the columns of web-mixing matrices which governs the
cancellation of the leading subdivergences between diagrams in the web. Our
results, when combined with parallel developments in the evaluation of
kinematic integrals, offer new insights into the all-order structure of
infrared singularities in non-Abelian gauge theories.
