The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

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 occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

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 number of mid points of increasing subsequences of length 3 in a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$. The generating function is given by [1].

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 big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.