Crossings and Nestings for Arc-Coloured Permutations and Automation

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.

Crossings and Nestings in Colored Set Partitions

“…In Section 5, we briefly survey several colored variations of these results. Our main finding is that many colored objects−such as matchings [7], permutations [6,37], and tangled diagrams [8]−can be realized as colored set partitions Λ satisfying certain conditions on min(Λ) and max(Λ); see Propositions 5.6, 5.12, and 5.16. The techniques we used to prove Theorem 1.4 can therefore also be used to easily derive the symmetric joint distribution of crossing and nesting numbers in these cases.…”

“…Several variations of the results in [10] have appeared for other objects besides set partitions for which natural concepts of crossings and nestings exists (see, e.g., [6,7,8,12,20,21,37]). In Section 5, we briefly survey several colored variations of these results.…”

“…In this final section we discuss a few variations of set partitions with natural notions of crossings, nestings, and colorings. Our discussion here is partially expository, surveying results from [6,7,8,9,37]. In each case our noncrossing and nonnesting colored objects are in bijection with certain classes of noncrossing and nonnesting colored set partitions.…”

“…Corteel [11] first introduced crossings and nestings for permutations (i.e., bijections [n] → [n]). Burill, Mishna, and Post [6] extended Corteel's notion to defines k-crossings and k-nestings in permutations, and Yen [37] has recently considered such crossings and nestings in colored permutations.…”

“…In other words, the number of partitions of [n] which are j-noncrossing and k-nonnesting is equal to the number of those which are k-noncrossing and j-nonnesting [10, Theorem 1.1]. The results of [10] have been reinterpreted and extended in a number of ways; we mention without hope of being comprehensive the papers [5,6,8,12,17,18,20,21,33,34,37]. Most recently, Chen and Guo [7] have generalized the equidistribution of crossings and nestings to colored complete matchings.…”