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Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

Authors
  • Foissy, Loïc
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
Jan 17, 2018
Volume
10
Issue
1
Pages
27–63
Identifiers
DOI: 10.1515/apam-2016-0051
Source
De Gruyter
Keywords
License
Yellow

Abstract

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism with values in ℚ ⁢ [ X ] \mathbb{Q}[X] . We deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber’s formula. We also give non-commutative versions of these results, where polynomials are replaced by packed words. We obtain, in particular, a non-commutative duality principle.

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