The rank function of a positroid and non-crossing partitions

TL;DR

This paper establishes a method to compute the rank of any set in a positroid directly from its decorated permutation using non-crossing partitions, linking combinatorial objects with geometric properties.

Contribution

It introduces a novel approach to determine set ranks in positroids via decorated permutations and non-crossing partitions, enhancing combinatorial understanding.

Findings

Rank of a set in a positroid can be computed from decorated permutation.

Non-crossing partitions provide a combinatorial tool for positroid analysis.

Connects positroid bases with non-crossing partitions and decorated permutations.

Abstract

A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.