The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

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.

The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that $$\alpha\pi\alpha^{-1} = \pi^2.$$

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 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]].