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 blocks with odd minimum. See [[St000746]] for the analogous statistic on perfect matchings.

The number of even descents of a permutation.

The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.

Lusztig's a-function for the symmetric group. Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$ by the Robinson-Schensted correspondence and $\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableau. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$. See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.

The Ulam distance of a permutation to the identity permutation. This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$. In other words, this statistic plus [[St000062]] equals $n$.

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