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

We introduce three definitions of q-analogs of Motzkin numbers and illustrate some combinatorial interpretations of these q-numbers. We relate the first class of q-numbers to the generating function for steep parallelogram polyominoes 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 counts the steep staircase polyominoes according to their area, while the third one enumerates the inversions of steep Dyck words. These enumerations allow us to illustrate various techniques of counting and q-counting. (C) 1998 Elsevier Science B.V. All rights reserved.