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 $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

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 number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$. We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.

The number of inversions of the third entry of a permutation. This is, for a permutation $\pi$ of length $n$, $$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.