On comparability of bigrassmannian permutations

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.