The number of bonds in a permutation. For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.

The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.

The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.

The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.

The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.

The Ulam distance of a permutation to the identity permutation. This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$. In other words, this statistic plus [[St000062]] equals $n$.

The Colin de Verdière graph invariant.

The number of natural descents for set-valued two row standard Young tableaux. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let $j$ be smallest integer such that $2j$ is a down step. Then $k$ is a natural descent if * $k-2\ge j$ and positions 2(k-1)-1,2(k-1) are a valley i.e. [0,1], or * $k-2\ge j$ and positions 2(k-1)-1,2(k-1) are a peak i.e. [1,0], or * $k-1\ge j$ and positions 2(k-1),2k-1,2k form [0,1,1], or * $k=j$ and positions 2k-1,2k are double down i.e. [0,0], or * $k < j$ and positions 2k-1,2k are a valley i.e. [0,1].