On the volume of a certain polytope

Let n greater than or equal to 2 be an integer and consider the set T-n of n x n permutation matrices pi for which pi(ij) = 0 for j greater than or equal to i+2. We study the convex hull P-n of T-n, a polytope of dimension ((n)(2)). We provide evidence for several conjectures involving P-n, including Conjecture 1: Let v(n) denote the minimum volume of a simplex with vertices in the affine lattice spanned by T-n. Then the volume of P-n is v(n) times the product [GRAPHICS] of the first n - 1 Catalan numbers. We also give a related result on the Ehrhart polynomial of P-n.