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},{2,3}} such that 1,2 are minimal.

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 number of inversions 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 $i$. A pair $(i,j)$ is an inversion of the word $w$ if $w_i > w_j$.

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.

The number of admissible inversions of a permutation. Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$. An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions: $1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.