The number of coloured trees such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of unlabelled trees on $n$ vertices, [[oeis:A000055]], whereas the value on the partition $(1^n)$ is the number of labelled trees [[oeis:A000272]].

The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.

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 minimal number of edges to add or remove to make a graph regular.

The minimal number of edges to add or remove to make a graph vertex transitive. A graph is vertex transitive if for any two edges there is an automorphism that maps one vertex to the other.