Linear operators preserving column majorization of $(0, 1)$-vectors

The paper provides a characterization of linear operators preserving column majorization of $(0, 1)$-vectors. In addition, such operators are characterized explicitly in the case where they are given by special matrices, namely, $(\pm 1)$-matrices of order not exceeding $10$ or $(0, \pm 1)$-matrices of order not exceeding $5$. A number of related combinatorial-matrix-theory results are also proved.