Explicit Expressions for the First 20 Moments of the Area Under Dyck and Motzkin Paths
AJ Bu, Shalosh B. Ekhad, and Doron Zeilberger
TL;DR
This paper derives explicit formulas for the first 20 moments of the area under Dyck and Motzkin paths, confirming their convergence to Brownian Excursion moments through elementary methods.
Contribution
It provides explicit sequences for moments of Dyck and Motzkin path areas, extending previous generating function results to actual moment formulas.
Findings
Explicit expressions for the first 20 moments of path areas
Confirmation of convergence to Brownian Excursion moments
Elementary derivation of limit behaviors
Abstract
Starting from AJ Bu's recent article that computed explicit expressions for the GENERATING functions of sums of powers of areas under Dyck and Motzkin paths, we deduce from them explicit expressions for the actual sequences. This enables taking the limits of the scaled moments and confirming, in an entirely elementary way, that they tend to those of the area under Brownian Excursion (up to any specified moment).
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Geometric and Algebraic Topology · Quasicrystal Structures and Properties