Contraction and restriction of positroids in terms of decorated permutations

The rank function of a positroid and non-crossing partitions

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.

Dyck Paths and Positroids from Unit Interval Orders

It is well known that the number of non-isomorphic unit interval orders on $[n]$ equals the $n$-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on $[n]$ naturally induces a rank $n$ positroid on $[2n]$. We call the positroids produced in this fashion unit interval positroids. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a $2n$-cycle encoding a Dyck path of length $2n$. We also provide recipes to read the decorated permutation of a unit interval po…

The facets of the matroid polytope and the independent set polytope of a positroid

A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.

Quotients of uniform positroids

Flag matroids are a rich family of Coxeter matroids that can be characterized using pairs of matroids that form a quotient. We consider a class of matroids called positroids, introduced by Postnikov, and utilize their combinatorial representations to explore characterizations of flag positroids. Given a uniform positroid, we give a purely combinatorial characterization of a family of positroids that form quotients with it. We state this in terms of their associated decorated permutations. In proving our characterization we also fully describe the circuits of this family.

稀疏铺砌正拟阵（Sparse Paving Positroids）

Sparse Paving Positroids

本研究利用Le-图、装饰排列与格拉斯曼项链分类稀疏铺砌正拟阵，完成其计数并建立与黄金比例相关序列、卢卡斯数的联系。

一致拟阵的positroid商的刻画

Characterizing positroid quotients of uniform matroids

本文研究两步flag positroid，完整刻画了包含一致拟阵的情形，证明Grassmann项链包含关系对一般flag positroid不充分，并解决了装饰排列通过循环移位相关的猜想。