University of Queensland Centre for Discrete Mathematics and Computing
Australasian Journal of Combinatorics
Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) 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), respectively), the number of BG permutations weakly below (weakly above, respectively) a ﬁxed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn.
We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β 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 β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain.