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A NOTE ON THE GESSEL NUMBERS
Keywords:
Gessel numbers, Catalan numbers, central binomial coefficient, lattice pathsDOI:
https://doi.org/10.17654/0972555524013Abstract
The Gessel number $P(n, r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r, n+r-1)$ that never touch any of the points from the set $\left\{(x, x) \in \mathbb{Z}^2: x \geq r\right\}$. In this paper, we use combinatorial arguments to derive a recurrence relation between $P(n, r)$ and $P(n-1, r+1)$. Also, we give a new proof for a well-known closed formula for $P(n, r)$. Moreover, a new combinatorial interpretation for the Gessel numbers is presented.
Received: December 15, 2023
Accepted: March 11, 2024
References
I. M. Gessel, Super ballot numbers, J. Symbolic Comput. 14 (1992), 179-194.
V. J. W. Guo, Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun, Electron. J. Combin. 21 (2014), Paper 2.54, 13 pp.
T. Koshy, Catalan Numbers with Applications, Oxford University Press, 2009.
C. Krattenhaler, Lattice Path Enumeration, arxiv.org/pdf/1503.05930.pdf.
J. Mikic, On certain sums divisible by the central binomial coefficient, J. Integer Seq. 23 (2020), Article 20.1.6, 22 pp.
R. P. Stanley, Bijective proof problems, 2009.
R. P. Stanley, Catalan Numbers, Cambridge University Press, Cambridge, 2015.
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