The number of inversions 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], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

The Z-index of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The Z-index of $w$ equals $$ \sum_{i < j} w_{i,j}, $$ where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.

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 lcb 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 '''lcb''' (left-closer-bigger) 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 dimension exponent of a set partition. This is $$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$ where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$. It is thus equal to the difference [[St000728]] - [[St000211]]. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.

The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.