  We compare the Hilbert series approach with explicit constructions of
higher-order Lagrangians for the $O(N)$ nonlinear sigma model. We use the
Hilbert series to find the number and type of operators up to mass dimension
16, for spacetime dimension $D$ up to 12 and $N$ up to 12, and further classify
the operators into spacetime parity and parity of the internal symmetry group
$O(N)$. The explicit construction of operators is done up to mass dimension 12
for both parities even and dimension 10 for the other three cases. The results
of the two methods are in full agreement. This provides evidence for the
Hilbert series conjecture regarding co-closed but not co-exact $k$-forms, which
takes into account the integration-by-parts relations.
