The sum of the minimal distances to a greater element. Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is $$ \sum_{i=1}^n \min_d(\pi_{i-d}>\pi_i\text{ or }\pi_{i+d}>\pi_i) $$ This statistic appears in [1]. The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently coincides with [2], which satisfies the functional equation $$ (x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x = 0. $$

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

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

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 largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The pebbling number of a connected graph.