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For a $d$-dimensional lattice polytope $P$ and positive integer $n$, Ehrhart reciprocity stateswith relative interior when needed. Negative input means polynomial evaluation, not a negative dilation count. For the unit square, $(1-n{)}^2=(n-1{)}^2$ counts the interior grid points of its $n$th dilation.^[ref]

Matthias Beck coauthored a text on integer-point enumeration in polyhedra with Sinai Robins. It connects generating functions, Ehrhart polynomials and Ehrhart reciprocity. Counting lattice points in dilated shapes thereby reveals geometric information such as volume, while reciprocity relates counts in the interior to counts in the closed shape.^[ref]

Sinai Robins coauthored a text on integer-point enumeration with Matthias Beck. Their exposition develops Ehrhart polynomials, generating functions and Ehrhart reciprocity for polyhedra. The subject gives a precise bridge between discrete counting and continuous geometry: the leading growth of lattice-point counts records volume.^[ref]