The dimension of the adjoint representation of the Lie group of given type.

The value of the complete homogeneous symmetric function evaluated at 1. The statistic is $h_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=x_2=\dotsb=x_k$, where $\lambda$ has $k$ parts.

The number of transitive factorisations of a permutation of given cycle type into star transpositions. Let $\pi$ be a permutation of cycle type $\lambda\vdash n$ and let $r=n + \ell(\lambda) - 2$. A minimal factorization of $\pi$ into star transpositions is an $r$-tuple of transpositions $(1, a_1)\dots(1, a_r)$ whose product (in this order) equals $\pi$. The number of such factorizations equals [1] $$ \frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}. $$

The Ramsey number of a graph. This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]

Gives the number of generalised tilting modules of the corresponding LNakayama algebra.

The product of the hook lengths of the integer partition. Consider the Ferrers diagram associated with the integer partition. For each cell in the diagram, drawn using the English convention, consider its ''hook'': the cell itself, all cells in the same row to the right and all cells in the same column below. The ''hook length of a cell'' is the number of cells in the hook of a cell. This statistic is the product of the hook lengths of all cells in the partition. Let $H_\lambda$ denote this product, then the number of standard Young tableaux of shape $\lambda$, (traditionally denoted $f^\lambda$) equals $n! / H_\lambda$. Therefore, it is consistent to set the product of the hook lengths of the empty partition equal to $1$.

The sum of the hook lengths of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the sum of all hook lengths of a partition.

The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra.