The number of supergreedy linear extensions of a poset.
A linear extension of a poset P with elements $\{x_1,\dots,x_n\}$ is supergreedy, if it can be obtained by the following algorithm:

• Step 1. Choose a minimal element $x_1$.
• Step 2. Suppose $X=\{x_1,\dots,x_i\}$ have been chosen, let $M$ be the set of minimal elements of $P\setminus X$. If there is an element of $M$ which covers an element $x_j$ in $X$, then let $x_{i+1}$ be one of these such that $j$ is maximal; otherwise, choose $x_{i+1}$ to be any element of $M$.

This statistic records the number of supergreedy linear extensions.

Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$

Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.

Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.