The number of inversions of a standard tableau. Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.

The weighted size of a partition. Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is $$\sum_{i=0}^m i \cdot \lambda_i.$$ This is also the sum of the leg lengths of the cells in $\lambda$, or $$ \sum_i \binom{\lambda^{\prime}_i}{2} $$ where $\lambda^{\prime}$ is the conjugate partition of $\lambda$. This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].

The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. Alternatively, remark that the monomials of the polynomial $\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with $n$ east steps: the exponent of $z_i$ is the number of north steps before the $i$-th east step, see [2]. Thus, this statistic records the coefficients of the monomials. A formula for the coefficient of $z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]: $$ c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}. $$ This polynomial arises in a partial symmetrization process as follows, see [1]. For $w\in\frak{S}_n$, let $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Furthermore, let $$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$ Then $\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$.

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

The number of inversions of a binary word.

The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.

The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.

The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$