With Tlist = [$t_1\, t_2\,..\.\,t_N$], the command GenerateSymmetricTensors(Tlist) will generate a list of symmetric tensors by forming all possible r-fold tensor products $t_{i_1}\otimes t_{i_2}..\.\otimes t_{i_r}$ and symmetrizing the result.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form GenerateSymmetricTensors(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-GenerateSymmetricTensors.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[w\,x\,y\,z\right]\,M\right)\:$

$\mathrm{T1}≔\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$\mathrm{T1}:=\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}\right]$

$\mathrm{L1}≔\mathrm{GenerateSymmetricTensors}\left(\mathrm{T1}\,2\right)$

$\mathrm{L1}:=\left[\mathrm{dx}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}\mathrm{dy}\mathrm{dx}\,\frac{1}{2}\mathrm{dx}\mathrm{dz}\+\frac{1}{2}\mathrm{dz}\mathrm{dx}\,\mathrm{dy}\mathrm{dy}\,\frac{1}{2}\mathrm{dy}\mathrm{dz}\+\frac{1}{2}\mathrm{dz}\mathrm{dy}\,\mathrm{dz}\mathrm{dz}\right]$

$\mathrm{nops}\left(\mathrm{L1}\right)$

$6$