A new statistic on Dyck paths for counting 3-dimensional Catalan words
Kassie Archer, Christina Gravies
TL;DR
This paper introduces a new statistic on Dyck paths to count 3-dimensional Catalan words, providing formulas involving Motzkin numbers for specific values, including primes.
Contribution
It defines a novel statistic on Dyck paths and derives enumeration formulas for 3D Catalan words based on this statistic, linking to Motzkin numbers.
Findings
Formulas for counting Dyck paths with the statistic equal to specific values.
Enumeration results for prime values of the statistic.
Connections established between Dyck paths, Motzkin numbers, and Catalan words.
Abstract
A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path , a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly . In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Bayesian Methods and Mixture Models · Advanced Mathematical Identities