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.

Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The number of big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.