Name
Abstract | ||
|---|---|---|
We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first relates the rational generating function σv+K(x) := Σm∈(v+K)∩ZdXm, where K is a rational cone and v ∈Rd, with σ-v+K°(1/x). The second theorem asserts that the generating function 1+Σn≥1LP(n)tn of the Ehrhart quasi-polynomial LP(n) := #(nP∩Zd) of a rational polytope P can be written as a rational function vP(t)/(1-t)dimP+1 with nonnegative numerator vP. The third theorem asserts that if P ⊆ Q, then vP ≤ vQ. Our proofs are based on elementary counting afforded by irrational decompositions of rational polyhedra. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1016/j.ejc.2005.06.003 | Eur. J. Comb. |
Keywords | Field | DocType |
rational polyhedron,rational generating function,nonnegative numerator vp,rational cone,ehrhart quasi-polynomial lp,generating function,irrational decomposition,rational function vp,rational polytope,integer point,irrational proof,rational function | Integer,Discrete mathematics,Generating function,Combinatorics,Polyhedron,Irrational number,Mathematical proof,Polytope,Rational function,Fraction (mathematics),Mathematics | Journal |
Volume | Issue | ISSN |
28 | 1 | European Journal of Combinatorics 28, no. 1 (2007), 403-409 |
Citations | PageRank | References |
12 | 1.51 | 1 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Matthias Beck | 1 | 54 | 10.27 |
| Frank Sottile | 2 | 26 | 5.10 |