Geometric properties of matrices induced by pattern avoidance

The notion of submatrix avoidance in polyominoes has recently been introduced in [2] with the aim of extending most of the concepts and properties concerning pattern avoiding permutations to the setting of polyominoes. In this paper we use submatrix avoidance to describe families of polyominoes which, in the literature, are usually defined by means of the geometric constraints of convexity, k-convexity, and directedness. To reach this goal, we provide an extension of the notion of pattern in a polyomino, by introducing generalized polyomino patterns. In the second part of the paper, we tackle the same problem in the context of discrete sets, which can be naturally regarded as binary matrices. In this case, we consider two types of geometric constraints: convexity and directedness, and we study how these constraints can be imposed on matrices by using submatrix avoidance.