The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].

The sorting index of a permutation. The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols. Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra. See theorem 5.8. in the reference for a motivation.

The number of visible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

The sum of the descent differences of a permutations. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by $$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$

The major index of a permutation. This is the sum of the positions of its descents, $$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$ Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$. A statistic equidistributed with the major index is called '''Mahonian statistic'''.