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.

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 number of two-by-two squares inside a skew partition. This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.

The number of simple modules with projective dimension two in the incidence algebra of the poset.

The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.

The weighted size of a partition. Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is $$\sum_{i=0}^m i \cdot \lambda_i.$$ This is also the sum of the leg lengths of the cells in $\lambda$, or $$ \sum_i \binom{\lambda^{\prime}_i}{2} $$ where $\lambda^{\prime}$ is the conjugate partition of $\lambda$. This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].

The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.