Algebraic aspects of some Riordan arrays related to binary words avoiding a pattern

We consider some Riordan arrays related to binary words avoiding a pattern p, which can be easily studied by means of an A-matrix rather than their A-sequence. Both concepts allow us to define every element as a linear combination of other elements in the array; the A-sequence is unique and corresponds to a linear dependence from the previous row. The A-matrix is not unique and corresponds to a linear dependence from several previous rows. However, for the problems considered in the present paper, we show that the A-matrix approach is more convenient. We provide explicit algebraic generating functions for these Riordan arrays and obtain many statistics on the corresponding languages. We thus obtain a deeper insight of the languages L[p] of binary words avoiding p having a number of 0s less or equal to the number of 1s.