The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.

The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.

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 boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.

The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. The top of a module is the cokernel of the inclusion of the radical of the module into the module. For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.

The independence gap of a graph. This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph. In particular, this statistic is $0$ for well covered graphs