Enumeration of pattern-avoiding (0,1)-matrices and their symmetry classes

Recently, Brualdi and Cao studied Ik[jls-end-space/]-avoiding (0,1)[jls-end-space/]-matrices by decomposing them into zigzag paths and proved that the maximum number of 1's in such a matrix is given by an exact formula. We further study the structure of maximal Ik[jls-end-space/]-avoiding (0,1)[jls-end-space/]-matrices (IAMs) by interpreting them as families of non-intersecting lattice paths on the square lattice. Using this perspective, we establish a bijection showing that IAMs are equinumerous with plane partitions of a certain size. Moreover, we classify all ten symmetry classes of IAMs under the action of the dihedral group of order 8 and show that the enumeration formulas for these classes are given by simple product formulas. Extending this approach to skew shapes, we derive a conceptual formula for enumerating maximal Ik[jls-end-space/]-avoiding (0,1)[jls-end-space/]-fillings of skew shapes.