The size of the centralizer of any permutation of given cycle type.
The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$:
$$C_g = \{h \in G : hgh^{-1} = g\}.$$
Its size thus depends only on the conjugacy class of $g$.
The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is
$$|C| = \Pi j^{a_j} a_j!$$
For example, for any permutation with cycle type $\lambda = (3,2,2,1)$,
$$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$
There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.

The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.