Journal of Integer Sequences, Vol. 25 (2022), Article 22.7.8

A Motzkin-Inspired Bijection

Peter Matsakis and Sam Vandervelde
Proof School
San Francisco, CA 94103


In this paper we briefly revisit Motzkin numbers before noting a novel manifestation of Motzkin numbers among certain fault-free tableaux. We next describe two other instances involving tableaux in which Motzkin numbers arise and present a bijection between them, yielding a new proof of the fact that standard Young tableaux with three or fewer rows are enumerated by the Motzkin numbers. We then extend the method of bijection to prove a new result that generalizes the original observation to larger tableaux. This approach motivates a natural way to generalize the Motzkin numbers, namely, as the sequences that result from fixing the height of the grid and letting the width vary. Finally, we note a few properties of these numbers and state two conjectures about their asymptotics.

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(Concerned with sequence A001006.)

Received February 3 2022; revised version received August 1 2022. Published in Journal of Integer Sequences, October 6 2022.

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