  Let n >= 2 be an integer and consider the set T_n of n by n permutation
matrices pi for which pi_{ij}=0 for j>=i+2. In this paper we study the convex
hull of T_n, which we denote by P_n. P_n is a polytope of dimension
binom{n}{2}. Our main purpose is to provide evidence for the following
conjecture concerning its volume. 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 for i varying from 0 to n-2 of frac{1}{i+1}
binom{2i}{i}. That is, P_n is the product of v_n and the first n-1 Catalan
numbers.
  We also give a related result on the Ehrhart polynomial of P_n.
