Steep polyominoes, q-Motzkin numbers and q-Bessel functions

We introduce three deenitions of the q-analogs of Motzkin numbers and illustrate some combinatorial interpretations of these q-numbers. We relate the rst class of q-numbers to the steep parallelogram polyominoes' generating function according to their width, perimeter and area. We show that this generating function is the quotient of two q-Bessel functions. The second class of q-Motzkin numbers enumerates the inversions of steep Dyck words, while the third one counts the steep staircase polyominoes according to their area. These enumerations allow us to illustrate various techniques of counting and q-counting.