The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.

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 interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].

Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.