The sum of the number of descents and the number of recoils of a permutation. This statistic is the sum of [[St000021]] and [[St000354]].

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 posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].

The number of posets with the same zeta polynomial. The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset. See section 3.12 of [1]. Since $$ Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k, $$ where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.

The number of posets with combinatorially isomorphic order polytopes.

The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.