The length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]

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 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 distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].

The number of upper covers of a partition in dominance order.

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].

The number of up steps after the last double rise of a Dyck path.