The flag excedance statistic of a signed permutation. This is the number of negative entries plus twice the number of excedances of the signed permutation. That is, $$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$ where $$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$ $$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$ It has the same distribution as the flag descent statistic.

The flag descent of a signed permutation. $$ fdes(\sigma) = 2 \lvert \{ i \in [n-1] \mid \sigma(i) > \sigma(i+1) \} \rvert + \chi( \sigma(1) < 0 ) $$ It has the same distribution as the flag excedance statistic.

The number of flag weak exceedances of a signed permutation. This is the number of negative entries plus twice the number of weak exceedances of the signed permutation.

The position of the last up step in a Dyck path.

The sum of the minimal distances to a greater element. Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is $$ \sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i) $$ This statistic appears in [1]. The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently coincides with [2], which satisfies the functional equation $$ (x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0. $$

The number of cuts of a poset. A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.

The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].