The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.

The number of occurrences of the pattern 31-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.

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 dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.