  We compute the type (maximum linearization) of the well partial order of
bounded lower sets in $\mathbb{N}^m$, ordered under inclusion, and find it is
$\omega^{\omega^{m-1}}$. Moreover we compute the type of the set of all lower
sets in $\mathbb{N}^m$, a topic studied by Aschenbrenner and Pong, and find
that it is equal to \[ \omega^{\sum_{k=1}^{m} \omega^{m-k}\binom{m}{k-1} }+ 1.
\] As a consequence we deduce corresponding bounds on effectively given
sequences of monomial ideals in $F[X,Y]$ where $F$ is a field.
