The 1-Color Problem and the Brylawski Model

In discrete tomography, the 1-color problem consists in determining the existence of a binary matrix with row and column sums equal to some given input values arranged in two vectors. These two vectors are said to be compatible if the associated 1-color problem has at least a solution. Here, we start from a vector of projections, and we define an algorithm to compute all the vectors compatible with it, then we show how to arrange them in a partial order structure, and we point out some of its combinatorial properties. Finally, we prove that this poset is a sublattice of the Brylawski lattice too, and we check some common properties.