Ascending runs in permutations and valued Dyck paths

Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, Matteo Silimbani

Abstract


We define a bijection between permutations and valued Dyck paths, namely, Dyck paths whose odd vertices are labelled with an integer that does not exceed their height. This map allows us to characterize the set of permutations avoiding the pattern 132 as the preimage of the set of Dyck paths with minimal labeling. Moreover, exploiting this bijection we associate to the set of n-permutations a polynomial that generalizes at the same time Eulerian polynomials, Motzkin numbers, super-Catalan numbers, little Schröder numbers, and other combinatorial sequences. Lastly, we determine the Hankel transform of the sequence of such polynomials.


Keywords


Permutation, Dyck path, pattern avoidance, Hankel transform

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DOI: https://doi.org/10.26493/1855-3974.1679.ad3

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications