The number of adjacent cycles of a permutation. This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).

The number of occurrences of the pattern 1324 in a permutation. There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that $$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$

The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. Given a partition $\lambda\vdash n$, let $\alpha(\lambda)$ be the partition given by the lengths of the antidiagonals of the Ferrers diagram of $\lambda$. Then, the value of the statistic on $\mu$ is the number of times $\mu$ appears in the multiset $\{\{\alpha(\lambda)\mid \lambda\vdash n\}\}$.

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.