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

Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.

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

The number of flag weak exceedances of a signed permutation. This is the number of negative entries plus twice the number of weak exceedances of the signed permutation.

The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.

The length of the longest palindromic prefix of a binary word.

The length of the longest palindromic factor beginning with a one of a binary word.

The number of times a permutation switches from increasing to decreasing or decreasing to increasing. This is the same as the number of inner peaks plus the number of inner valleys and called alternating runs in [2]

The length of the exterior of a permutation. The '''exterior''' of a permutation is the longest proper prefix that is also a suffix, when viewed as a pattern. In other words, the length of the exterior of a permutation $\sigma$ of length $n$ is the largest $i < n$ such that the first $i$ entries of $\sigma$ are in the same relative order as the last $i$ entries of $\sigma$.