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SIMPLICES FOR NUMERAL SYSTEMS
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2019 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 371, no 3, p. 2089-2107Article in journal (Refereed) Published
Abstract [en]

The family of lattice simplices in 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*-polynomials. Here we show, via an association with numeral systems, that such simplices yield h*-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 nth place value of a positional numeral system. We then observe that their h*-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*-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 >= 2, and prove that the associated h*-polynomials are real-rooted and unimodal for r >= 2 and n >= 1.

Place, publisher, year, edition, pages
AMER MATHEMATICAL SOC , 2019. Vol. 371, no 3, p. 2089-2107
Keywords [en]
Simplex, Ehrhart, weighted projective space, real-rooted, unimodal, symmetric, Eulerian polynomial, binomial coefficients, numeral system, factoradics, binary
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-241305DOI: 10.1090/tran/7424ISI: 000454641300021Scopus ID: 2-s2.0-85062075859OAI: oai:DiVA.org:kth-241305DiVA, id: diva2:1283101
Note

QC 20190128

Available from: 2019-01-28 Created: 2019-01-28 Last updated: 2019-03-18Bibliographically approved

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  • apa
  • harvard1
  • ieee
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  • Other style
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  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
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  • asciidoc
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