Motzkin and Catalan Tunnel Polynomials
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo
Dipartimento di Matematica
Università di Bologna
SSPG "M. Marinelli"
We define sequences MTn and CTn of polynomials associated with Motzkin
and Catalan paths, respectively. We show that these polynomials satisfy
recurrence relations 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 Schröder 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.
Full version: pdf,
(Concerned with sequences
Received March 28 2018; revised versions received September 7 2018; December 3 2018.
Published in Journal of Integer Sequences, December 5 2018.
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