Sparse Paving Positroids

TL;DR

The paper addresses the classification and enumeration of sparse paving positroids, linking positroid combinatorics to the totally nonnegative Grassmannian via Le-diagrams, Grassmann necklaces, and decorated permutations. It proves that for rank $2\le k\le n-2$, sparse paving positroids correspond to non-adjacent subsets of $[n]$ (via Grassmann necklaces) and to decorated permutations obtained by applying non-adjacent adjacent transpositions to the top uniform permutation. It provides a Γ-diagram realization that encodes the same condition and enables counting: the number of sparse paving positroids on $[n]$ (for $2\le k\le n-2$) equals the Lucas numbers $L_n$, independent of $k$, with a Fibonacci-type recurrence and relation to the golden ratio. This yields exact enumeration and ties to classical sequences, enriching understanding of positroid structure in the totally nonnegative Grassmannian and paving matroid theory.

Abstract

Using Postnikov's Le-diagrams, decorated permutations, and Grassmann necklaces, we classify which positroids are sparse paving matroids. This allows us to enumerate sparse paving positroids, making connections to a known sequence involving the golden ratio and to the Lucas numbers.

Figures (3)

• Figure 1: The numbering of the cells for $k = 4$ and $n = 10$.
• Figure 2: On the left, we have the $\raisebox{\depth}{[-1]{$\Gamma$}}$-diagram $D_{4,10}(A)$, where $A=\{1,3,10\}$. On the right, we have $D_{4,10}(A)$ where $A=\{2,3,9\}$.
• Figure 3: The $\raisebox{\depth}{[-1]{$\Gamma$}}$-diagram $D_{4,12}(\{6\})$ and an illustration that $C_{4,12}^{(6)}=\{6,7,8,9\}$ is not a basis due to the path from $1$ to $9$ colliding with another path above sink $8$.

Theorems & Definitions (18)

• Lemma 2.1
• Definition 3.1: Pos
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Definition 4.1: Decorated Permutation Pos
• Theorem 4.2
• proof
• Definition 5.1: $\raisebox{\depth}{[-1]{$\Gamma$}}$-diagram Pos