The major index of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The major index of $w$ is the sum of the positions $i$ such that $w_i > w_{i+1}$.

The disorder of a permutation. Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process. For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.