The number of minimal length factorizations of a permutation into star transpositions. For a permutation $\pi\in\mathfrak S_n$ a minimal length factorization into star transpositions is a factorization of the form $$\pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n,$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$ and $k$ is minimal. [1, lem.2.1] shows that the minimal length of such a factorization is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$, see [[St001077]]. [2, cor.2] shows that the number of such minimal factorizations is $$\frac{(n+m-2(k+1))!}{(n-k)!}\ell_1\cdots\ell_m,$$ where $\ell_1,\dots,\ell_m$ is the cycle type of $\pi$ and $k$ is the number of fixed point different from $1$.

The cube of the number of standard Young tableaux with shape given by the partition.

The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The number of self-evacuating tableaux of given shape. This is the same as the number of standard domino tableaux of the given shape.

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.