Simple marked mesh patterns

In this paper we begin the first systematic study of distributions of simple marked mesh patterns. Mesh patterns were introduced recently by Br"and'en and Claesson in connection with permutation statistics. We provide explicit generating functions in several general cases, and develop recursions to compute the numbers in question in some other cases. Certain $q$-analogues are discussed. Moreover, we consider two modifications of the notion of a marked mesh pattern and provide enumerative results for them.