The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].

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

The charge of a standard tableau.

The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.

The number of alignments in a perfect matching. An alignment is a pair of edges $(i,j)$, $(k,l)$ such that $i < j < k < l$. Since any two edges in a perfect matching are either nesting ([[St000041]]), crossing ([[St000042]]) or form an alignment, the sum of these numbers in a perfect matching with $n$ edges is $\binom{n}{2}$.

The Rajchgot index of a permutation. The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.

The number of occurrences of the pattern {{1},{2}} in a set partition.

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 size of a partition. This statistic is the constant statistic of the level sets.