| Two permutations of [n] := {lcub}1, 2,..., n{rcub} are comparable in the Bruhat order if one can be obtained from the other by a sequence of transpositions decreasing the number of inversions. We show that the total number of pairs of permutations (pi,sigma) with pi ≤ sigma is of order (n!)2/n 2 at most. Equivalently, if pi,sigma are chosen uniformly at random and independently of each other, then P(pi ≤ sigma) is of order n-2 at most. By a direct probabilistic argument we prove P(pi ≤ sigma) is of order (0.708)n at least, so that there is currently a wide qualitative gap between the upper and lower bounds.; Next, emboldened by a connection with Ferrers diagrams and plane partitions implicit in Bressoud's book [13], we return to the Bruhat order upper bound and show that for n-permutations pi1,..., pi r selected independently and uniformly at random, Pp1≤&cdots;≤p r=On-r r-1, thus providing an extension of our results for pairs of permutations to chains of length r > 2.; Turning to the related weak order " ⪯ "---when only adjacent transpositions are admissible---we use a non-inversion set criterion to prove that P*n := P(pi ⪯ sigma) is submultiplicative, thus showing existence of rho = lim P*nn . We demonstrate that rho is 0.362 at most. Moreover, we prove the lower bound i=1n (H(i)/i) for P*n , where H(i) := j=1i 1/j. In light of numerical experiments, we conjecture that for each order the upper bounds for permutation-pairs are qualitatively close to the actual behavior. We believe that extensions to r-chains similar to that for the Bruhat order upper bound can be made for our other bounds in each order, and are presently working in this direction.; Finally, the weak order poset happens to be a lattice, and we study some properties of its infimums and supremums. Namely, we prove that the number of r-tuples (pi1,..., pir) of n-permutations with minimal infimum, 12··· n, asymptotically equals -n!r h'rz* z* n+1,r≥2,n→ infinity.1 Here, z* = z *(r) ∈ (1, 2) is the unique (positive) root of the equation hrz:= j≥0-1 jj!r zj=0 within the disk |z| ≤ 2. Moreover, (1) is also the asymptotic number of r-tuples with maximal supremum, n(n - 1)···1. |
