The number of distinct Laplacian eigenvalues of a graph.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Half the size of the symmetry class of a permutation. The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]). This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.

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 pancake length of a permutation. This is the minimal number of pancake moves needed to generate a permutation where a pancake move is a reversal of a prefix in a permutation.

The number of prefix or suffix reversals needed to sort a permutation. A prefix reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of an initial subsequence of the form $\operatorname{reversal}_{j}(\pi) = [\pi_j,\ldots,\pi_{1},\pi_{j+1},\ldots,\pi_n]$ for $1 < j \leq n$. Final reversals are defined analogously. This statistic is then given by the minimal number of initial or final reversals needed to sort a permutation.

The number of non-attacking neighbors of a permutation. For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$. Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.