Abstract
Symmetric joint distribution between crossings and nestings was established in several combinatorial objects. Recently, Marberg extended Chen and Guo's result on coloured matchings to coloured set partitions following a multi-dimensional generalization of the bijection and enumerative methods from Chen, Deng, Du, Stanley, and Yan. We complete the study for arc-coloured permutations by establishing symmetric joint distribution for crossings and nestings and by showing that the ordinary generating functions for $j$-noncrossing, $k$-nonnesting, $r$-coloured permutations according to size $n$ are rational functions. Finally, we automate the generation of these rational functions and analyse the first $70$ series.
Highlights
Crossing and nesting statistics have intrigued combinatorialists for many decades
We show that the ordinary generating functions for j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions, and automate the generation of these rational functions
With automated generation of their rational series according to crossing and nesting statistics, random generation algorithms like Boltzmann sampling [9] can be applied for the investigation of the distribution of such structures modelled by coloured permutations
Summary
Crossing and nesting statistics have intrigued combinatorialists for many decades. For example, it is well known that. On coloured set partitions, we combine two theorems of [14] to establish symmetric joint the electronic journal of combinatorics 22(1) (2015), #P1.14 distribution of crossing and nesting statistics for arc-coloured permutations. We show that the ordinary generating functions for j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions, and automate the generation of these rational functions. R-coloured permutations may be a natural model for the study of genome rearrangement problems, in particular, for tracking different types of distance metrics [10]. With automated generation of their rational series according to crossing and nesting statistics, random generation algorithms like Boltzmann sampling [9] can be applied for the investigation of the distribution of such structures modelled by coloured permutations
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