The size of the center of the Weyl group of a finite Cartan type.

The greatest common divisor of the parts of the partition.

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.

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 sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. For example, $p_{22} = s_{1111} - s_{211} + 2s_{22} - s_{31} + s_4$, so the statistic on the partition $22$ is 2. This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2] For a permutation $\pi$ of given cycle type, this is also the number of permutations whose square equals $\pi$. [2]

The Grundy value of the game of creating an independent set in a graph. Two players alternately add a vertex to an initially empty set, which is not adjacent to any of the vertices it already contains. Alternatively, the game can be described as starting with a graph, the players remove vertices together with their neighbors, until the graph is empty.

Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. Here the Coxeter polynomial is by definition the Coxeter polynomial of the corresponding LNakayama algebra.

The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.

The number of finite solvable groups that are realised by the given partition over the complex numbers. A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers. The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.

The number of finite groups that are realised by the given partition over the complex numbers. A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.