The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].

The number of maximal chains of minimal length in a poset.

The number of greedy linear extensions of a poset. A linear extension of a poset $P$ with elements $\{x_1,\dots,x_n\}$ is greedy, 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. If there is at least one minimal element of $P\setminus X$ which is greater than $x_i$ then choose $x_{i+1}$ to be any such minimal element; otherwise, choose $x_{i+1}$ to be any minimal element of $P\setminus X$. This statistic records the number of greedy linear extensions.

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.