The number of pairs whose larger element is at most one more than half the size of the perfect matching. Under the bijection between perfect matchings of $\{1,\dots,2n\}$ and rooted unordered binary trees with $n+1$ labelled leaves described in Example 5.2.6 of [1], this is the number of nodes having two leaves as children. The number of perfect matchings of $\{1,\dots,2n\}$ with $j$ such pairs, computed in [2], is $$\frac{(2j-3)!! (n+1)!}{2^j j!}\binom{n-1}{2j-2}.$$

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.

The number of crossings of a set partition. This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.

The breadth of a permutation. According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$ According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$. By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$. By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this. According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.