The dimension of a set partition. This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks. A slightly shifted definition of the dimension is [[St000572]].

The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$\sum_v (j_v-i_v)/2.$$

The reduced reflection length of the permutation. Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is $$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$ In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].

The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.

The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). In symbols, for a permutation $\pi$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$. Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.

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 $$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$