The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

The number of parts of an integer partition that are at least two.

The number of runs of ones in a binary word.

The number of descents of a binary word.

The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.

The number of rises of length 2 of a Dyck path. This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].

The number of rises of length at least 2 of a Dyck path.

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not 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 $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$. See also [[St000213]] and [[St000119]].