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In this thesis, we consider the problem of characterizing and enumeratingsets of polyominoes described in terms of some constraints, defined either byconvexity or by pattern containment. We are interested in a well known subclassof convex polyominoes, the k-convex polyominoes for which the enumerationaccording to the semi-perimeter is known only for k=1,2. We obtain, from arecursive decomposition, the generating function of the class of k-convexparallelogram polyominoes, which turns out to be rational. Noting that thisgenerating function can be expressed in terms of the Fibonacci polynomials, wedescribe a bijection between the class of k-parallelogram polyominoes and theclass of planted planar trees having height less than k+3. In the second partof the thesis we examine the notion of pattern avoidance, which has beenextensively studied for permutations. We introduce the concept of patternavoidance in the context of matrices, more precisely permutation matrices andpolyomino matrices. We present definitions analogous to those given forpermutations and in particular we define polyomino classes, i.e. sets downwardclosed with respect to the containment relation. So, the study of the old andnew properties of the redefined sets of objects has not only becomeinteresting, but it has also suggested the study of the associated poset. Inboth approaches our results can be used to treat open problems related topolyominoes as well as other combinatorial objects.

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints