The number of tolerances of a finite lattice. Let $L$ be a lattice. A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join. Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$. The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].

The normalized Knill dimension of a graph. The Knill dimension [1], [2] is a rational number associated with a graph as follows: for the empty graph $\dim(G) = -1$. For a graph with non-empty vertex set $V$, it is $\dim(G) = 1 + \frac{1}{|V|}\sum_{v\in V}\dim(N_v)$, where $N_v$ is the subgraph of $G$ induced by the set of neighbours of $v$. Conjecturally, the least common multiple of the denominators of all graphs is the order of the alternating group [[oeis:A001710]]. Thus, the normalized Knill dimension is the Knill dimension multiplied with the number.

The composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

The number of non-convex subsets of vertices in a graph. A set of vertices $U$ is convex, if for any two vertices $u,v\in U$, all vertices on any shortest path connecting $u$ and $v$ are also in $U$.