The number of Coxeter elements in the Weyl group of a finite Cartan type. This is, the elements that are conjugate to the product of the simple generators in any order, or, equivalently, the elements that admit a primitive $h$-th root of unity as an eigenvalue where $h$ is the Coxeter number.

The largest coefficient in the Poincaré polynomial of the Weyl group of given Cartan type. The Poincaré polynomial of a Weyl group $W$ is $$ \sum_{w\in W} q^{\ell(w)} = \prod_i [d_i]_q, $$ where $\ell$ denotes the Coxeter length, $d_1,\dots$ are the degrees (or exponents) of $W$ and $[n]_q=1 +\dots+q^{n-1}$ is the $q$-integer. Thus, this statistic records the frequency of the most common length in the group.

The number of elements of maximal order in the Weyl group of a finite Cartan type. For the symmetric group $\mathfrak S_n$ this is [[OEIS:A074859]].

The maximum cut size of a graph. A '''cut''' is a set of edges which connect different sides of a vertex partition $V = A \sqcup B$.

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$, where $k$ is the number of parts of $\lambda$. An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.

The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.

The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.

The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [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 sum of the squares of the parts of a partition.