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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Simplices for numeral systems


Author: Liam Solus
Journal: Trans. Amer. Math. Soc. 371 (2019), 2089-2107
MSC (2010): Primary 52B05, 52B20; Secondary 05A05, 05A10
DOI: https://doi.org/10.1090/tran/7424
Published electronically: October 1, 2018
MathSciNet review: 3894046
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Abstract: The family of lattice simplices in $ \mathbb{R}^n$ formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative algebraic geometry and mirror symmetry. From this perspective, it is useful to have formulae for their discrete volumes via Ehrhart $ h^\ast $-polynomials. Here we show, via an association with numeral systems, that such simplices yield $ h^\ast $-polynomials with properties that are also desirable from a combinatorial perspective. First, we identify $ n$-simplices in this family that associate via their normalized volume to the $ n$th place value of a positional numeral system. We then observe that their $ h^\ast $-polynomials admit combinatorial formula via descent-like statistics on the numeral strings encoding the nonnegative integers within the system. With these methods, we recover ubiquitous $ h^\ast $-polynomials including the Eulerian polynomials and the binomial coefficients arising from the factoradic and binary numeral systems, respectively. We generalize the binary case to base-$ r$ numeral systems for all $ r\geq 2$, and prove that the associated $ h^\ast $-polynomials are real-rooted and unimodal for $ r\geq 2$ and $ n\geq 1$.


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Additional Information

Liam Solus
Affiliation: Matematik, KTH, SE-100 44 Stockholm, Sweden
Email: solus@kth.se

DOI: https://doi.org/10.1090/tran/7424
Keywords: Simplex, Ehrhart, weighted projective space, real-rooted, unimodal, symmetric, Eulerian polynomial, binomial coefficients, numeral system, factoradics, binary
Received by editor(s): June 1, 2017
Received by editor(s) in revised form: September 16, 2017
Published electronically: October 1, 2018
Additional Notes: The author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship (DMS - 1606407).
Article copyright: © Copyright 2018 American Mathematical Society