The Dynkin index of the Lie algebra of given type. This is the greatest common divisor of the Dynkin indices of the representations of the Lie algebra. It is computed in [2, prop.2.6].

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$

Number of simple reflexive modules that are 2-stable reflexive. See Definition 3.1. in the reference for the definition of 2-stable reflexive.

The number of graphs with the same chromatic polynomial.

The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.

The number of graphs with the same symmetric edge polytope as the given graph. The symmetric edge polytope of a graph on $n$ vertices is the polytope in $\mathbb R^n$ defined as the convex hull of $e_i - e_j$ and $e_j - e_i$ for each edge $(i, j)$, where $e_1,\dots, e_n$ denotes the standard basis.

The genus of a graph. This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.

The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path.

The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.