The command GeneratingFunction(P) returns  the generating function of P This is a multivariate rational function with a number of variables equal to the dimension of P. If P is bounded, evaluating all the variables of GeneratingFunction(P) produces the number of integer points of P.

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$$\mathrm{with}\left(\mathrm{ExampleSets}\right)\:$$\mathrm{with}\left(\mathrm{ValuesUnderConstraints}\right)\:$

$\mathrm{ps}≔\mathrm{Tetrahedron}\left(\right)\;$$\mathrm{PolyhedralSets}:-\mathrm{Plot}\left(\mathrm{ps}\right)$

$\mathrm{ps}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_1-x_2-x_3\le 1\,-x_1+x_2+x_3\le 1\,x_1-x_2+x_3\le 1\,x_1+x_2-x_3\le 1\right]\end{array}$

$\mathrm{gps}≔\mathrm{GeneratingFunction}\left(\mathrm{ps}\right)$

$\mathrm{gps}≔\frac{x_1^2{x}_2^2{x}_3^2+x_1^2{x}_2{x}_3+x_3{x}_1{x}_2^2+x_2{x}_1{x}_3^2+x_1{x}_2{x}_3+x_1^2+x_1{x}_2+x_1{x}_3+x_2^2+x_2{x}_3+x_3^2}{x_1{x}_2{x}_3}$

$\mathrm{NumberOfIntegerPoints}\left(\mathrm{ps}\right)$

$11$

$\mathrm{ps}≔\mathrm{TruncatedOctahedron}\left(\right)\;$$\mathrm{PolyhedralSets}:-\mathrm{Plot}\left(\mathrm{ps}\right)$

$\mathrm{ps}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,x_3\le 1\,-x_2\le 1\,x_2\le 1\,-x_1-x_2-x_3\le \frac{3}{2}\,-x_2+x_3-x_1\le \frac{3}{2}\,-x_1\le 1\,-x_3+x_2-x_1\le \frac{3}{2}\,-x_1+x_2+x_3\le \frac{3}{2}\,x_1-x_3-x_2\le \frac{3}{2}\,\mathrm{and\; 4\; more\; constraints}\right]\end{array}$

$\mathrm{gps}≔\mathrm{GeneratingFunction}\left(\mathrm{ps}\right)$

$\mathrm{gps}≔\frac{x_2{x}_3{x}_1^2+x_2^2{x}_1{x}_3+x_3^2{x}_1{x}_2+x_2{x}_1{x}_3+x_1{x}_2+x_1{x}_3+x_2{x}_3}{x_2{x}_1{x}_3}$

$\mathrm{NumberOfIntegerPoints}\left(\mathrm{ps}\right)$

$7$

$\mathrm{NIP}≔\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,1\le j\,i\le n\,j\le n\right]\,\left[i\,j\right]\,\left[n\right]\right)\:$$\mathrm{ToPiecewise}\left(\mathrm{NIP}\right)$

$\mathrm{ToPiecewise}\left(\left[⟨\text{value}\left\{n^2\right\}\text{when}\left[0\le -2+n\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[n-1=0\right]⟩\right]\right)$

$\mathrm{NIP}≔\mathrm{NumberOfIntegerPoints}\left(\left[-i\le -1\,i\le n\,-j\le -1\,j-i\le 0\right]\,\left[i\,j\right]\,\left[n\right]\right)\:$$\mathrm{ToPiecewise}\left(\mathrm{NIP}\right)$

$\mathrm{ToPiecewise}\left(\left[⟨\text{value}\left\{\frac{1}{2}{n}^2+\frac{1}{2}n\right\}\text{when}\left[0\le -2+n\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[n-1=0\right]⟩\right]\right)$

$\mathrm{NIP}≔\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,j\le n\,i\le m\,3i\le 5j\right]\,\left[i\,j\right]\,\left[m\,n\right]\right)\:$$\mathrm{ToPiecewise}\left(\mathrm{NIP}\right)$

$\mathrm{ToPiecewise}\left(\left[⟨\text{value}\left\{⟨\text{one of the polynomials}\left(\begin{array}{c}0\\ 0\\ -\frac{2}{5}\\ -\frac{1}{5}\\ -\frac{2}{5}\end{array}\right)\text{depending on the value of}m\text{modulo}5⟩+nm-\frac{3m^2}{10}+\frac{3m}{10}\right\}\text{when}\left[0\le m-2\,0\le 5n-7\,0\le 5n-3m-1\right]⟩\,⟨\text{value}\left\{n\right\}\text{when}\left[m-1=0\,0\le 5n-4\right]⟩\,⟨\text{value}\left\{\frac{5n^2}{6}+\frac{n}{2}+⟨\text{one of the polynomials}\left(\begin{array}{c}0\\ -\frac{1}{3}\\ -\frac{1}{3}\end{array}\right)\text{depending on the value of}n\text{modulo}3⟩\right\}\text{when}\left[3m-5n=0\,0\le 5n-4\right]⟩\,⟨\text{value}\left\{\frac{5n^2}{6}+\frac{n}{2}+⟨\text{one of the polynomials}\left(\begin{array}{c}0\\ -\frac{1}{3}\\ -\frac{1}{3}\end{array}\right)\text{depending on the value of}n\text{modulo}3⟩\right\}\text{when}\left[0\le 5n-4\,0\le -5n+3m-1\right]⟩\right]\right)$

$\mathrm{NIP}≔\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,i\le n\,i\le m\,1\le j\,j\le p\right]\,\left[i\,j\right]\,\left[m\,n\,p\right]\right)\;$$\mathrm{ToPiecewise}\left(\mathrm{NIP}\right)$

$\mathrm{NIP}≔\left[⟨\text{value}\left\{pn\right\}\text{when}\left[m-n=0\,0\le -2+n\,0\le -2+p\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[m-1=0\,n-1=0\,0\le -2+p\right]⟩\,⟨\text{value}\left\{n\right\}\text{when}\left[p-1=0\,m-n=0\,0\le -2+n\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[m-1=0\,n-1=0\,p-1=0\right]⟩\,⟨\text{value}\left\{pn\right\}\text{when}\left[0\le -2+n\,0\le -2+p\,0\le m-n-1\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[n-1=0\,0\le -2+p\,0\le m-2\right]⟩\,⟨\text{value}\left\{n\right\}\text{when}\left[p-1=0\,0\le -2+n\,0\le m-n-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[n-1=0\,p-1=0\,0\le m-2\right]⟩\,⟨\text{value}\left\{pm\right\}\text{when}\left[0\le -2+p\,0\le m-2\,0\le n-3\,0\le -m+n-1\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[m-1=0\,0\le -2+n\,0\le -2+p\right]⟩\,⟨\text{value}\left\{m\right\}\text{when}\left[p-1=0\,0\le m-2\,0\le n-3\,0\le -m+n-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[m-1=0\,p-1=0\,0\le -2+n\right]⟩\right]$

$\mathrm{ToPiecewise}\left(\left[⟨\text{value}\left\{pn\right\}\text{when}\left[m-n=0\,0\le -2+n\,0\le -2+p\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[m-1=0\,n-1=0\,0\le -2+p\right]⟩\,⟨\text{value}\left\{n\right\}\text{when}\left[p-1=0\,m-n=0\,0\le -2+n\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[m-1=0\,n-1=0\,p-1=0\right]⟩\,⟨\text{value}\left\{pn\right\}\text{when}\left[0\le -2+n\,0\le -2+p\,0\le m-n-1\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[n-1=0\,0\le -2+p\,0\le m-2\right]⟩\,⟨\text{value}\left\{n\right\}\text{when}\left[p-1=0\,0\le -2+n\,0\le m-n-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[n-1=0\,p-1=0\,0\le m-2\right]⟩\,⟨\text{value}\left\{pm\right\}\text{when}\left[0\le -2+p\,0\le m-2\,0\le n-3\,0\le -m+n-1\right]⟩\,⟨\text{value}\left\{p\right\}\text{when}\left[m-1=0\,0\le -2+n\,0\le -2+p\right]⟩\,⟨\text{value}\left\{m\right\}\text{when}\left[p-1=0\,0\le m-2\,0\le n-3\,0\le -m+n-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[m-1=0\,p-1=0\,0\le -2+n\right]⟩\right]\right)$

Rui-Juan Jing, Yuzhuo Lei, Christopher F. S. Maligec, Marc Moreno Maza: " Counting the Integer Points of Parametric Polytopes: A Maple Implementation." Proceedings of Computer Algebra in Scientific Computing - 26th International Workshop (CASC) 2024: 140-160, Lecture Notes in Computer Science, vol. 14938, Springer. ##

The PolyhedralSets[GeneratingFunction] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.