Let k, n denote two positive integers and consider the family of the polytopes defined as the convex hull of pairs of the form (Y, h) where Y is a 0/1-matrix with k rows, n columns, containing exactly one nonzero coefficient per column, and where h stands for the smallest index of a nonzero row of Y. These polytopes and some variants naturally emerge in formulations of different classical combinatorial optimization problems such as minimum makespan scheduling and minimum span frequency assignment. In this paper, we provide complete formulations for these polytopes and show the associated separation problem can be solved in polynomial time. The complete formulations in the original space of variables generally contain an exponential number of inequalities. Alternative extended compact formulations are also presented