Crossings and Nestings for Arc-Coloured Permutations

Abstract: 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.