The charge of a standard tableau.

The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.

The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.

The shifted natural comajor index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural comajor index of a tableau of shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.

The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.

The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].

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.

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'''.

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 bounce statistic of a Dyck path. The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$. The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$. In particular, the bounce statistics of $D$ and $D'$ coincide.