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 invariant subsets of size 2 when acting with a permutation of given cycle type.

The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

The sum of the greatest common divisors of all pairs of parts.

The number of occurrences of the vincular pattern |123 in a permutation. This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.

The number of occurrences of the pattern 12-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $12\!\!-\!\!3$.

The number of occurrences of the vincular pattern |312 in a permutation. This is the number of occurrences of the pattern $(3,1,2)$, such that the letter matched by $3$ is the first entry of the permutation.

The number of occurrences of the vincular pattern |321 in a permutation. This is the number of occurrences of the pattern $(3,2,1)$, such that the letter matched by $3$ is the first entry of the permutation.

The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. Let $D$ be a Dyck path, and let $P$ be the noncrossing set partition obtained by applying [[Mp00138]]. For each pair of singleton blocks $\{a\}, \{b\}$, let $P'$ be the set partition obtained from $P$ by merging the two blocks. This statistic enumerates the number of (unordered) pairs of singleton blocks such that $P'$ is noncrossing.

The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that $$\alpha\pi\alpha^{-1} = \pi^2.$$