The number of occurrences of the pattern 132 in a permutation.

The number of occurrences of the pattern 132 or of the pattern 231 in a permutation.

The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.

The number of occurrences of the pattern 132 or of the pattern 321 in a permutation.

The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.

A descent variant minus the number of inversions. This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots. For $w \in W$ this statistic is then $$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$ where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length. It was shown in [1], that for irreducible groups, it holds that $$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$ where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice. For a permutation $\sigma \in S_n$, this becomes $$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$

The number of alternating sign matrices whose left key is the permutation. The left key of an alternating sign matrix was defined by Lascoux in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

The number of reduced Kogan faces with the permutation as type. This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.

The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.

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: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$