Locally Invariant Positions of \((0,1)\) Matrices

Let \( {R} = (r_1, r_2, \ldots, r_m)\) and \( {S} = (s_1, s_2, \ldots, s_n)\) be nonnegative integral vectors. Denote by \( {A}( {R}, {S})\) the class of \((0,1)\) matrices with row sum vector \( {R}\) and column sum vector \( {S}\). We study a generalization of invariant positions called locally invariant positions of a class \( {A}( {R}, {S})\). For a normalized class, locally invariant positions have in common with invariant positions the property that they lie above and to the left of some simple rook path through the set of positions.