### Obtaining the poset of the Catalan monoid

The poset $P_n$ is defined as the poset consisting of subsets of ~~${1,...,n }$ ~~ { 1,...,n } where for two subsets $X \leq Y$ if and only if $X$ and $Y$ have the same cardinality and if ~~$X= ~~X= {x_1 < ... < x_k ~~}$ ~~} and ~~$Y= ~~Y= {y_1 < ... < y_k ~~}$ ~~} we have $x_i \leq y_i$ for $i=1,...,k$. See for example https://arxiv.org/pdf/1806.06531.pdf .

My question is whether the is an easy way to obtain this poset for a given $n$ with Sage?