The number of nestings of a perfect matching. This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.

The number of crossings of a perfect matching. This is the number of pairs of edges $((a,b),(c,d))$ such that $a\le c\le b\le d$, i.e., the edges $(a,b)$ and $(c,d)$ cross when drawing the perfect matching as a chord diagram. The generating function for perfect matchings $M$ of $\{1,\dots,2n\}$ according to the number of crossings $\textrm{cr}(M)$ is given by the Touchard-Riordan formula ([2], [4], a bijective proof is given in [7]): $$ \sum_{M} q^{\textrm{cr}(M)} = \frac{1}{(1-q)^n} \sum_{k=0}^n\left(\binom{2n}{n-k}-\binom{2n}{n-k-1}\right)(-1)^k q^{\binom{k+1}{2}} $$

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 nestings 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 rcs statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.

The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image. Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).

The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.

Sum of the odd parts of a partition.

The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.