On k-crossings and k-nestings of permutations

TL;DR

This paper introduces the concepts of k-crossings and k-nestings in permutations, demonstrating their symmetric distribution and establishing equal counts for k-noncrossing and k-nonnesting permutations, along with enumerative results.

Contribution

It defines k-crossings and k-nestings in permutations and proves their symmetric distribution and equal enumeration for k-noncrossing and k-nonnesting permutations.

Findings

Crossing and nesting numbers have a symmetric joint distribution.

Number of k-noncrossing permutations equals the number of k-nonnesting permutations.

Provides enumerative results for k-noncrossing permutations.

Abstract

We introduce k-crossings and k-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of k-noncrossing permutations is equal to the number of k-nonnesting permutations. We also provide some enumerative results for k-noncrossing permutations for some values of k.