  The equidistribution of many crossing and nesting statistics exists in
several combinatorial objects like matchings, set partitions, permutations, and
embedded labelled graphs. The involutions switching nesting and crossing
numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du,
Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post
involved passing through tableau-like objects. Recently, Chen and Guo for
matchings, and Marberg for set partitions extended the result to coloured arc
annotated diagrams. We prove that symmetric joint distribution continues to
hold for arc-coloured permutations. As in Marberg's recent work, but through a
different interpretation, we also conclude that the ordinary generating
functions for all $j$-noncrossing, $k$-nonnesting, $r$-coloured permutations
according to size $n$ are rational functions. We use the interpretation to
automate the generation of these rational series for both noncrossing and
nonnesting coloured set partitions and permutations.
