The number of pop-stack-sorts needed to sort a permutation. The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order. A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.

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

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

The number of cycles in the cycle decomposition of a permutation.