{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:42Z","timestamp":1753893822368,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A rational polytope is the convex hull of a finite set of points in ${\\Bbb R}^d$ with rational coordinates. Given a rational polytope ${\\cal P} \\subseteq {\\Bbb R}^d$, Ehrhart proved that, for $t\\in{\\Bbb Z}_{\\ge 0}$, the function $\\#(t{\\cal P} \\cap {\\Bbb Z}^d)$ agrees with a quasi-polynomial $L_{\\cal P}(t)$, called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart\u2013Macdonald theorem on reciprocity.<\/jats:p>","DOI":"10.37236\/340","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:13:43Z","timestamp":1578716023000},"source":"Crossref","is-referenced-by-count":2,"title":["A Finite Calculus Approach to Ehrhart Polynomials"],"prefix":"10.37236","volume":"17","author":[{"given":"Steven V.","family":"Sam","sequence":"first","affiliation":[]},{"given":"Kevin M.","family":"Woods","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,4,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r68\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r68\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:53:51Z","timestamp":1579305231000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r68"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/340","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,4,30]]},"article-number":"R68"}}