The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

The "bounce" of a permutation.

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

The bounce statistic of a Dyck path. The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$. The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$. In particular, the bounce statistics of $D$ and $D'$ coincide.

The size of the left subtree of a binary tree.

The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b

The number of ascent tops in the permutation such that all smaller elements appear before.

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 touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.