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The number of gridpoints on hyperplane sections of the $ d$-dimensional cube


Author: Ulrich Abel
Journal: Proc. Amer. Math. Soc. 146 (2018), 5349-5355
MSC (2010): Primary 52B20; Secondary 05A15
DOI: https://doi.org/10.1090/proc/14233
Published electronically: September 4, 2018
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Abstract: We deduce a formula for the exact number of gridpoints (i.e., elements of $ \mathbb{Z}^{d}$) in the extended $ d$-dimensional cube $ nC_{d}=\left [ -n,+n \right ] ^{d}$ on intersecting hyperplanes. In the special case of the hyperplanes $ \{ x\in \mathbb{R}^{d}\mid x_{1}+\cdots +x_{d}\linebreak =b\} $, $ b\in \mathbb{Z}$, these numbers can be written as a finite sum involving products of certain binomial coefficients. Furthermore, we consider the limit as $ n$ tends to infinity which can be expressed in terms of Euler-Frobenius numbers. Finally, we state a conjecture on the asymptotic behaviour of this limit as the dimension $ d$ tends to infinity.


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Additional Information

Ulrich Abel
Affiliation: Department MND, Technische Hochschule Mittelhessen, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany
Email: ulrich.abel@mnd.thm.de

DOI: https://doi.org/10.1090/proc/14233
Received by editor(s): February 27, 2017
Published electronically: September 4, 2018
Dedicated: In loving memory of my dear wife Anke $($1967–2018$)$
Communicated by: Mourad Ismail
Article copyright: © Copyright 2018 American Mathematical Society

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