The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by $$\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)$$

The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.

The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].

The number of edges of a graph.

The number of Bruhat lower covers of a permutation. This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Number of indecomposable injective modules with projective dimension 2.

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

The reversal length of a permutation. A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$. This statistic is then given by the minimal number of reversals needed to sort a permutation. The reversal distance between two permutations plays an important role in studying DNA structures.