Motzkin and Catalan Tunnel Polynomials

We define sequences MTn and CTn of polynomials associated with Motzkin and Catalan paths, respectively. We show that these polynomials satisfy recurrence rela- tions similar to the one satisfied by Motzkin and Catalan numbers. We study in detail many different specializations of these polynomials, which turn out to be sequences of great interest in combinatorics, such as the Schr ̈oder numbers, Fibonacci numbers, q-Catalan polynomials, and Narayana polynomials. We show a connection between the polynomials CTn and the family of binary trees, which allows us to find another specialization for our polynomials in term of path length in these trees. In the last section we extend the previous results to partial and free Motzkin paths.