The natural major 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 major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.

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 sum of the positions of the strong records of an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. This statistic is the sum of the positions of the strong records.

The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. The total number of words with letter multiplicities given by an integer partition is [[St000048]]. For example, there are twelve words with letters $0,0,1,2$ corresponding to the partition $[2,1,1]$. Two of these contain the pattern $132$: $0,0,2,1$ and $0,2,1,0$. Note that this statistic is not constant on compositions having the same parts. The number of words of length $n$ with letters in an alphabet of size $k$ avoiding the consecutive pattern $132$ is determined in [1].

Number of parabolic noncrossing partitions indexed by the composition. Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.

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 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.

The dinv of a Dyck path. Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]). The dinv statistic of $D$ is $$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$ Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by $$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$

The number of edges of a graph.

The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b