On comparability of bigrassmannian permutations

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

Convex (0,1)(0,1)-Matrices and Their Epitopes

We investigate $(0,1)$-matrices that are {\em convex}, which means that the ones are consecutive in every row and column. These matrices occur in discrete tomography. The notion of ranked essential sets, known for permutation matrices, is extended to convex sets. We show a number of results for the class \mc{C}(R,S) of convex matrices with given row and column sum vectors $R$ and $S$. Also, it is shown that the ranked essential set uniquely determines a matrix in \mc{C}(R,S)