#### Document Type

Article

#### Language

eng

#### Publication Date

2018

#### Publisher

University of Queensland Centre for Discrete Mathematics and Computing

#### Source Publication

Australasian Journal of Combinatorics

#### Source ISSN

1034-4942

#### Abstract

Let S_{n} and G_{n} denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (G_{n},≤) denote the Bruhat ordering permutation poset. We study the restricted poset (B_{n},≤), 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 ∈ B_{n}, and is used to compute the Mo¨bius function on any interval in B_{n}.

We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ B_{n}. 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/n^{3}. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain.

#### Recommended Citation

Engbers, John and Hammett, Adam, "On comparability of bigrassmannian permutations" (2018). *Mathematics, Statistics and Computer Science Faculty Research and Publications*. 581.

https://epublications.marquette.edu/mscs_fac/581

## Comments

Published version.

Australiasian Journal of Combinatorics, Vol. 71, No. 1, (2018): 121-152. Publisher link. © 2018 University of Queensland Centre for Discrete Mathematics and Computing. Used with permission.