$k$-distant crossings and nestings of matchings and partitions

TL;DR

This paper introduces k-distant crossings and nestings in matchings and partitions, demonstrating their symmetric distribution, analyzing their enumeration for small k, and connecting them to orthogonal polynomials and known combinatorial sequences.

Contribution

It extends the concepts of crossings and nestings by incorporating vertex distance, providing new symmetry results and enumerations for these generalized structures.

Findings

Joint distribution of k-distant crossings and nestings is symmetric.

k-distant noncrossing matchings and partitions are counted by known sequences.

Connections to orthogonal polynomials are established.

Abstract

We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of k-distant crossings and nestings is symmetric. We also study the numbers of k-distant noncrossing matchings and partitions for small k, which are counted by well-known sequences, as well as the orthogonal polynomials related to k-distant noncrossing matchings and partitions. We extend Chen et al.'s r-crossings and enhanced r-crossings.