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 minimal number of transpositions needed to sort a permutation in either direction. For a permutation $\sigma$, this is $\min\{ \operatorname{inv}(\sigma),\operatorname{inv}(\tau)\}$ where $\tau$ is the reverse permutation sending $i$ to $\sigma(n+1-i)$.

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.

The dimension of the maximal parabolic seaweed algebra corresponding to the partition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be two compositions of $n$. The corresponding seaweed algebra is the associative subalgebra of the algebra of $n\times n$ matrices which preserves the flags $$ \{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V $$ and $$ V=W_0\supset W_1\supset \cdots \supset W_t=\{0\}, $$ where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$. Thus, its dimension is $$ \frac{1}{2}\left(\sum a_i^2 + \sum b_i^2\right). $$ It is maximal parabolic if $b_1=n$.

The value of the forgotten symmetric functions when all variables set to 1. Let $f_\lambda(x)$ denote the forgotten symmetric functions. Then the statistic associated with $\lambda$, where $\lambda$ has $\ell$ parts, is $f_\lambda(1,1,\dotsc,1)$ where there are $\ell$ variables substituted by $1$.

Twice the mean value of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by $$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$ where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column.

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd. The case of an even minimum is [[St000621]].