The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.

The comajor index for set-valued two-row standard Young tableaux. The comajorindex is the sum $\sum_k (n+1-k)$ over all natural descents $k$. Bijections via bicolored Motzkin paths (with two restrictions, see [1]) give the following for Dyck paths. Let $j$ be smallest integer such that $2j$ is a down step. Then $k$ is a natural descent if * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a valley i.e. [0,1], or * $k-2\ge j$ and positions $2(k-1)-1,2(k-1)$ are a peak i.e. [1,0], or * $k-1\ge j$ and positions $2(k-1),2k-1,2k$ form [0,1,1], or * $k=j$ and positions $2k-1,2k$ are double down i.e. [0,0], or * $k < j$ and positions $2k-1,2k$ are a valley i.e. [0,1].

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 circled entries. asdasda An entry of a Gelfand-Tsetlin pattern is circled if $a_{i,j} = a_{i-1,j}$ (the northeast neighbor is the same).

The number of boxed entries. An entry of a Gelfand-Tsetlin pattern is boxed if $a_{i,j} = a_{i-1,j-1}$ (the northwest neighbor is the same).

The number of Dyck paths that are weakly above the Dyck path, except for the path itself.

The major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.

The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1]. This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2] Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.