On comparability of bigrassmannian permutations

Let Sn and Bn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Sn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), 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 β(ω) (α(ω) resp.), the number of BG permutations weakly below (weakly above resp.) a fixed ω ∈ Bn. We then turn to a probabilistic study of β = β(ω) (α = α(ω) resp.) for the uniformly random ω ∈ Bn. As a consequence of a product moment calculation, we show that α and β are equidistributed, and that β is of the same order as its expectation E [β] = |Bn+2| 10 = 1 10 ( n+ 3 3 ) , with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n, which we show converges to that of the random variable 1 2 L2L3(L2 + L3), where (L2, L3) are the respective second and third of four consecutive random lengths L1 = X, L2 = Y −X, L3 = Z − Y and L4 = 1− Z arising from rank-ordering three independent and uniformly random points from the unit interval 0 < X < Y < Z < 1. We use these results to show that the number of 2and 3-element multichains of BG permutations in Bruhat order are |Bn||Bn+2| 10 and (n + 4n+ 6)|Bn||Bn+3||Bn+6| 420(n+ 6)(n+ 7) , respectively. Thus, the restriction of the Bruhat order poset to BG permutations admits a proportion of comparable pairs (π, σ) with π ≤ σ close to 1 10 , and the fraction of triples (π, σ, τ) with π ≤ σ ≤ τ is close to 1 420 , both with error term of order n−1.