A variant of the major index of a set partition. For a set partition $P = B_1|\dots|B_k$ in canonical form (this is, each block is ordered increasingly and all blocks are ordered by their smallest element), one defined $\pi = \pi(P)$ to be the permutation obtained by writing the letters in all blocks as one-line notation and $\omega = \omega(P) = (\omega_1,\ldots,\omega_k)$ be to be the integer composition of the ordered block sizes. This statistic is then given in [1, (2.7)] by $$\operatorname{maj}(\pi) + \sum_{max\ B_i < min\ B_{i+1}} (\omega_1 + \cdots + \omega_i - i).$$

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