The number of prime labellings of a graph. A prime labelling of a graph is a bijective labelling of the vertices with the numbers $\{1,\dots, |V(G)|\}$ such that adjacent vertices have coprime labels.

The row sums of the character table of the symmetric group. Equivalently, this is the multiplicity of the irreducible representation corresponding to the given partition in the adjoint representation of the symmetric group.

Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].

The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.

The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.

The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. For example, $h_{11} = 2m_{11} + m_2$, so the statistic on the partition $11$ is 3.

The number of ordered refinements of an integer partition. This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.

The number of multiset partitions such that the multiplicities of elements are given by a partition. In particular, the value on the partition $(n)$ is the number of integer partitions of $n$, [[oeis:A000041]], whereas the value on the partition $(1^n)$ is the number of set partitions [[oeis:A006110]].

The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of partitions of $n$, whereas the value on the partition $(1^n)$ is the number of permutations.