The command NumberOfIntegerPoints(P) returns  the number of integer points of P, if P is bounded.

The command NumberOfIntegerPoints(PP,Vars,Params) returns the number of integer points of the parametric polyhedral set PP, with variables in Vars and parameters in Params, if PP is bounded.

By default, the value that is returned for the second calling sequence is a list of ValuesUnderConstraints objects. Using the ValuesUnderConstraints:-Eval command, one can evaluate this list to a list of sets of values, when specializing the parameters to a particular point. If you pass the output = piecewise option, then the output is a piecewise expression. This piecewise expression evaluates to a set of values, when specializing the parameters to a particular point. You can explicitly request the default output by passing the output = list option. In some circumstances (e.g. if there aren't any parameters), the ValuesUnderConstraints objects or piecewise expressions will be fully evaluated to a list of a set of a single integer (with output = list) or a set of a single integer (with output = piecewise).

The constraints in either the ValuesUnderConstraints objects or the piecewise expressions are systems of linear equations and non-strict linear inequalities in the parameters, while the values are polynomials or quasipolynomials in the parameters. A quasipolynomial is an object that assumes the value of one of several polynomials depending on the value of a parameter modulo an integer. The quasipolynomials will also evaluate to concrete values when one applies ValuesUnderConstraints:-Eval.

By default, any quasipolynomials are displayed in a format that explains to some extent what they represent. In some circumstances it may be better to show the quasipolynomials in a more compact way. This can be selected by using the compactdisplay option.

$\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)$

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

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

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

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

$7$

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

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

$\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,1\le j\,i\le n\,j\le n\right]\,\left[i\,j\right]\,\left[n\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{n^2\right\}& 0\le -2+n\\ \left\{1\right\}& n-1=0\end{array}\right.$

$\mathrm{NumberOfIntegerPoints}\left(\left[-i\le -1\,i\le n\,-j\le -1\,j-i\le 0\right]\,\left[i\,j\right]\,\left[n\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{\frac{1}{2}{n}^2+\frac{1}{2}n\right\}& 0\le -2+n\\ \left\{1\right\}& n-1=0\end{array}\right.$

$\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,j\le n\,i\le m\,3i\le 5j\right]\,\left[i\,j\right]\,\left[m\,n\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

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

$\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,j\le n\,i\le m\,3i\le 5j\right]\,\left[i\,j\right]\,\left[m\,n\right]\,\mathrm{output}=\mathrm{piecewise}\,\mathrm{compactdisplay}\right)$

$\left\{\begin{array}{cc}\left\{nm-\frac{3m^2}{10}+\mathrm{QuasiPolynomial}\left(\left[0\,0\,-\frac{2}{5}\,-\frac{1}{5}\,-\frac{2}{5}\right]\,m\,5\right)+\frac{3m}{10}\right\}& 0\le m-2\wedge 0\le 5n-7\wedge 0\le 5n-3m-1\\ \left\{n\right\}& m-1=0\wedge 0\le 5n-4\\ \left\{\mathrm{QuasiPolynomial}\left(\left[0\,-\frac{1}{3}\,-\frac{1}{3}\right]\,n\,3\right)+\frac{5n^2}{6}+\frac{n}{2}\right\}& 3m-5n=0\wedge 0\le 5n-4\\ \left\{\mathrm{QuasiPolynomial}\left(\left[0\,-\frac{1}{3}\,-\frac{1}{3}\right]\,n\,3\right)+\frac{5n^2}{6}+\frac{n}{2}\right\}& 0\le 5n-4\wedge 0\le -5n+3m-1\end{array}\right.$

$\mathrm{NumberOfIntegerPoints}\left(\left[1\le i\,i\le n\,i\le m\,1\le j\,j\le p\right]\,\left[i\,j\right]\,\left[m\,n\,p\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{pn\right\}& m-n=0\wedge 0\le -2+n\wedge 0\le -2+p\\ \left\{p\right\}& m-1=0\wedge n-1=0\wedge 0\le -2+p\\ \left\{n\right\}& p-1=0\wedge m-n=0\wedge 0\le -2+n\\ \left\{1\right\}& m-1=0\wedge n-1=0\wedge p-1=0\\ \left\{pn\right\}& 0\le -2+n\wedge 0\le -2+p\wedge 0\le m-n-1\\ \left\{p\right\}& n-1=0\wedge 0\le -2+p\wedge 0\le m-2\\ \left\{n\right\}& p-1=0\wedge 0\le -2+n\wedge 0\le m-n-1\\ \left\{1\right\}& n-1=0\wedge p-1=0\wedge 0\le m-2\\ \left\{pm\right\}& 0\le -2+p\wedge 0\le m-2\wedge 0\le n-3\wedge 0\le -m+n-1\\ \left\{p\right\}& m-1=0\wedge 0\le -2+n\wedge 0\le -2+p\\ \left\{m\right\}& p-1=0\wedge 0\le m-2\wedge 0\le n-3\wedge 0\le -m+n-1\\ \left\{1\right\}& m-1=0\wedge p-1=0\wedge 0\le -2+n\end{array}\right.$

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

Error, (in PolyhedralSets:-NumberOfIntegerPoints) polyhedral set must be bounded

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[NumberOfIntegerPoints] command was introduced in Maple 2025.

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