The pyramid weight of the Dyck path. The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path. Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.

The hull number of a graph. The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$. The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.

The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].

The monophonic hull number of a graph. The monophonic hull of a set of vertices $M$ of a graph $G$ is the set of vertices that lie on at least one induced path between vertices in $M$. The monophonic hull number is the size of the smallest set $M$ such that the monophonic hull of $M$ is all of $G$. For example, the monophonic hull number of a graph $G$ with $n$ vertices is $n$ if and only if $G$ is a disjoint union of complete graphs.

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].

The number of external nodes of a binary tree. That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].

The number of left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$. This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.

The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.