The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.

The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].

The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.

The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

The number of double deficiencies of a permutation. A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.

The number of 1/2-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] exhibit various posets that have a $1/2$-balanced pair.

The number of inversions between excedances and fixed points of a permutation. This is, $$\operatorname{iefp}(\sigma) = \#\{1 \leq i,j \leq n \mid i < j = \sigma(j) < \sigma(i) \}.$$

The number of invisible descents of a permutation. A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$. Thus, an invisible descent satisfies $\pi(i) > \pi(i+1) > i$.