The number of almost positive roots of a finite Cartan type. A root in the root system of a Cartan type is almost positive if it is either positive or simple negative. These are known to be in bijection with cluster variables in the cluster algebra of the given Cartan type, see [1]. This is also equal to the sum of the degrees of the fundamental invariants of the group.

The number of posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].

The number of edges of a graph.

The size of the centralizer of any permutation of given cycle type. The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$: $$C_g = \{h \in G : hgh^{-1} = g\}.$$ Its size thus depends only on the conjugacy class of $g$. The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is $$|C| = \Pi j^{a_j} a_j!$$ For example, for any permutation with cycle type $\lambda = (3,2,2,1)$, $$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$ There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.

The least common multiple of the parts of the partition.

The product of the parts of an integer partition.

The exponens consonantiae of a partition. This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].

The logarithm of the number of winning configurations of the lights out game on a graph. In the single player lamps out game, every vertex has two states, on or off. The player can toggle the state of a vertex, in which case all the neighbours of the vertex change state, too. The goal is to reach the configuration with all vertices off.

The maximum cut size of a graph. A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.

The number of non-isomorphic sublattices of a lattice.