A Motzkin-Inspired Bijection
Peter Matsakis and Sam Vandervelde
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.
Full version: pdf,
(Concerned with sequence
Received February 3 2022; revised version received August 1 2022.
Published in Journal of Integer Sequences,
October 6 2022.
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