The Dynkin index of the Lie algebra of given type. This is the greatest common divisor of the Dynkin indices of the representations of the Lie algebra. It is computed in [2, prop.2.6].

Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.

The number of invariant subsets of size 2 when acting with a permutation of given cycle type.

The sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1].

The multiplicity of the largest part of an integer partition.

The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

The number of parts from which one can substract 2 and still get an integer partition.

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a 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 $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

The smallest positive integer that does not appear twice in the partition.