A Note on the Gessel Numbers
Abstract
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 $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.12931
- arXiv:
- arXiv:2203.12931
- Bibcode:
- 2022arXiv220312931M
- Keywords:
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- Mathematics - Combinatorics;
- Primary 05A10;
- Secondary 11B65
- E-Print:
- 7 pages, no figures