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 indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

The number of occurrences of the vincular pattern |12-3 in a permutation. This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.