The command SumOfAllMonomials(x) returns the power series containing each monomial in the variables x with coefficient 1; that is, its homogeneous part of degree d is the sum of all monomials of degree d.

When using the MultivariatePowerSeries package, do not assign anything to the variables occurring in the power series, Puiseux series, and univariate polynomials over these series. If you do, you may see invalid results.

$\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right)\:$

$a≔\mathrm{SumOfAllMonomials}\left(\left[x\right]\right)$

$a≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{1-x}\mathrm{:}1+x+\mathrm{\dots }\right]$

$\mathrm{HomogeneousPart}\left(a\,2\right)$

$x^2$

$\mathrm{Truncate}\left(a\,4\right)$

$x^4+x^3+x^2+x+1$

$b≔\mathrm{SumOfAllMonomials}\left(\left[x\,y\,z\right]\right)$

$b≔\left[\mathrm{Pow\ⅇrS\ⅇri\ⅇs\; of}\frac{1}{\left(1-x\right)\left(1-y\right)\left(1-z\right)}\mathrm{:}1+x+y+z+\mathrm{\dots }\right]$

$\mathrm{HomogeneousPart}\left(b\,2\right)$

$x^2+yx+zx+y^2+zy+z^2$

$\mathrm{Truncate}\left(b\,3\right)$

$x^3+x^{2}y+x^{2}z+x{y}^2+xyz+x{z}^2+y^3+y^{2}z+y{z}^2+z^3+x^2+yx+zx+y^2+zy+z^2+x+y+z+1$

The MultivariatePowerSeries[SumOfAllMonomials] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.