Binary Pictures with Excluded Patterns Daniela Battaglino 1 , Andrea Frosini 2 , Veronica Guerrini 1 , Simone Rinaldi 1 , and Samanta Socci 1 1 Universit`a di Siena, Dipartimento di Matematica e Informatica, Pian dei Mantellini 44, 53100 Siena 2 Universit`a di Firenze, Dipartimento di Sistemi e Informatica, viale Morgagni 65, 50134 Firenze Abstract. The notion of a pattern within a binary picture (polyomino) has been introduced and studied in [3], and resembles the notion of pat- tern containment within permutations. The main goal of this paper is to extend the studies of [3] by adopting a more geometrical approach: we use the notion of pattern avoidance in order to recognize or describe families of polyominoes defined by means of geometrical constraints or combinatorial properties. Moreover, we extend the notion of pattern in a polyomino, by introducing generalized polyomino patterns, so that to be able to describe more families of polyominoes known in the literature. 1 Patterns in Binary Pictures and Polyomino Classes In recent years a considerable interest in the study of the notion of pattern within a combinatorial structure has grown. This kind of research started with patterns in permutations [12], while in the last few years it is being carried on in several directions. One of them is to define and study analogues of the concept of pattern in permutations in other combinatorial objects such as set partitions [11,14], words, trees [13]. The works [3,4] fit into this research line, in particular [4] introduces and studies the notion of pattern in finite binary pictures (specifically, in polyominoes). A finite binary picture is an m × n matrix of 0’s and 1’s. Intuitively speaking, 1’s correspond to black pixels (which constitute the image) and the 0’s corre- spond to white pixels (which form the background). Often, the studied images should fulfill several additional properties like symmetry, connectivity, or con- vexity. In particular, an image is connected if the set of black pixels is connected with respect to the edge-adjacency relation. A connected image is usually called a polyomino (see Figure 1). The work [3], from which we borrow most of the basic definitions and no- tations, uses an algebraic setting to provide a unified framework to describe and handle some families of binary pictures (in particular polyominoes), by the avoidance of patterns. Therefore, in order to fruitfully present our paper, we need to recall some definitions and the main results from [3]. Let M be the class of binary pictures (or matrices). We denote by  the usual subpicture (or submatrix) order on M, i.e. M ′  M if M ′ may be obtained from M by deleting any collection of rows and/or columns. E. Barcucci et al. (Eds.): DGCI 2014, LNCS 8668, pp. 25–38, 2014. c  Springer International Publishing Switzerland 2014