Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.8

Motzkin and Catalan Tunnel Polynomials

Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo
Dipartimento di Matematica
Università di Bologna
Bologna, 40126

Matteo Silimbani
SSPG "M. Marinelli"
Forlimpopoli, 47034


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.

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(Concerned with sequences A000045 A000108 A001006 A001263 A006318 A097610 A097860 A098978 A114583 A129181 A132893 A138157 A181371.)

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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