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Obtaining the poset of the Catalan monoid

The poset $P_n$ is defined as the poset consisting of subsets of ${1,...,n }$ where for two subsets $X \leq Y$ if and only if $X$ and $Y$ have the same cardinality and if $X= {x_1 < ... < x_k }$ and $Y= {y_1 < ... < y_k }$ 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?

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?