The number of up-down runs of a permutation. An '''up-down run''' of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$. For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and $643$.

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

The number of distinct Laplacian eigenvalues of a graph.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.

The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.