The sum of the sizes of the right subtrees of a binary tree. This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree. It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.

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 area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.

Gives the vector space dimension of the homomorphism space between J^2 and J^2.

The number of transpositions that are smaller or equal to a permutation in Bruhat order. A statistic is known to be '''smooth''' if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.

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 number of nestings of a perfect matching. This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.