Simplices for numeral systems
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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 $n\geq 1$.References
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Additional Information
- Liam Solus
- Affiliation: Matematik, KTH, SE-100 44 Stockholm, Sweden
- MR Author ID: 923057
- Email: solus@kth.se
- 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).
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3894046