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In this paper we present new results for the combinatorics of web diagramsand web worlds. These are discrete objects that arise in the physics ofcalculating scattering amplitudes in non-abelian gauge theories. Web-colouringand web-mixing matrices (collectively known as web matrices) are indexed byordered pairs of web-diagrams and contain information relating the number ofcolourings of the first web diagram that will produce the second diagram.  We introduce the black diamond product on power series and show how itdetermines the web-colouring matrix of disjoint web worlds. Furthermore, weshow that combining known physical results with the black diamond product givesa new technique for generating combinatorial identities. Due to the complicatedaction of the product on power series, the resulting identities appear highlynon-trivial.  We present two results to explain repeated entries that appear in the webmatrices. The first of these shows how diagonal web matrix entries will be thesame if the comparability graphs of their associated decomposition posets arethe same. The second result concerns general repeated entries in conjunctionwith a flipping operation on web diagrams.  We present a combinatorial proof of idempotency of the web-mixing matrices,previously established using physical arguments only. We also show how theentries of the square of the web-colouring matrix can be achieved by a lineartransformation that maps the standard basis for formal power series in onevariable to a sequence of polynomials. We look at one parameterized web worldthat is related to indecomposable permutations and show how determining theweb-colouring matrix entries in this case is equivalent to a combinatorics onwords problem.

Web matrices: structural properties and generating combinatorial identities