The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].

The depth index of a set partition. For a set partition $\Pi$ of $\{1,\dots,n\}$ with arcs $\mathcal A$, this is $$\sum_{i=1}^{|\mathcal A|} (n-i) - \sum_{j=1}^n depth(j) + \sum_{\alpha\in\mathcal A} depth(\alpha),$$ where the depth of an element $i$ is the number of arcs $(k,\ell)$ with $k < i < \ell$, and the depth of an arc $(i,j)$ is the number of arcs $(k,\ell)$ with $k < i$ and $j < \ell$.