On comparability of bigrassmannian permutations

TL;DR

This paper investigates the structure and properties of bigrassmannian permutations within the Bruhat order, providing criteria for comparability, enumerations of chains and elements, and probabilistic analysis of permutation statistics.

Contribution

It introduces a simple comparability criterion for bigrassmannian permutations and derives formulas for various order-related quantities, along with probabilistic results on permutation distributions.

Findings

The poset of bigrassmannian permutations is connected.

Formulas for the number of permutations below or above a fixed element.

Distribution of permutation statistics and probabilities of forming multichains.

Abstract

Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of ordinary and bigrassmannian (BG) permutations of order $n$, and let $(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study the restricted poset $(\mathfrak{B}_n,\leq)$, first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below $r$ fixed elements. It also quickly produces formulas for $\beta(\omega)$ ($\alpha(\omega)$ respectively), the number of BG permutations weakly below (weakly above respectively) a fixed $\omega\in\mathfrak{B}_n$, and is used to compute the M\"obius function on any interval in $\mathfrak{B}_n$. We then turn to a probabilistic study of $\beta=\beta(\omega)$ ($\alpha=\alpha(\omega)$ respectively) for the uniformly random $\omega\in\mathfrak{B}_n$. We show that $\alpha$ and $\beta$ are equidistributed, and that $\beta$ is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of $\beta/n^3$. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain.