Realizing Partitions Respecting Full and Partial Order Information

For n∈N, we consider the problem of partitioning the interval [0,n) into k subintervals of positive integer lengths ℓ1,…,ℓk such that the lengths satisfy a set of simple constraints of the form ℓi⋄ijℓj where ⋄ij is one of <, >, or =. In the full information case, ⋄ij is given for all 1⩽i,j⩽k. In the sequential information case, ⋄ij is given for all 1