The reversal length of a permutation. A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$. This statistic is then given by the minimal number of reversals needed to sort a permutation. The reversal distance between two permutations plays an important role in studying DNA structures.

The length of the longest partition in the vacillating tableau corresponding to a set partition. To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$. Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row. Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$. This statistic is the length of the longest partition on the diagonal of the diagram.

The largest even part of an integer partition.

The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. Following Stembridge [1, cor.4.7], the highest weight words indexing the irreducibles in $\mathfrak{gl_n}^{\otimes r}$ are ''staircase tableaux'' of length $2r$: sequences $(\gamma^{(0)},\dots,\gamma^{(2r)})$ of vectors in $\mathbb Z^n$ with decreasing entries, such that $\gamma^{(2i+1)}$ is obtained from $\gamma^{(2i)}$ by adding a unit vector and $\gamma^{(2i)}$ is obtained from $\gamma^{(2i-1)}$ by subtracting a unit vector. For $n=2$, the staircase tableaux whose final element is the zero vector are in natural correspondence with Dyck paths: adding the first or subtracting the second unit vector is translated to an up step, whereas adding the second or subtracting the first unit vector is translated to a down step. A Dyck path can be transformed into a ''bicoloured Motzkin path'' by replacing double up steps (double down, up-down, down-up steps) with up steps (down, coloured level, level steps). Note that the resulting path cannot have coloured level steps at height zero. In this context, say that a bicoloured Motzkin path has a $\mathfrak{gl}_2$-''descent'' between the following pairs of steps: * an up step followed by a level step * an up step followed by a down step, if the final height is not zero * a coloured level step followed by any non-coloured step. Then, conjecturally, the quasisymmetric expansion of the Frobenius character of the symmetric group $\mathfrak S_r$ acting on $\mathfrak{gl}_2^{\otimes r}$, is $$ \sum_M F_{Des(M)}, $$ where the sum is over all length $r$ prefixes of bicoloured Motzkin paths, $Des(M)$ is the set of indices of descents of the path $M$ and $F_D$ is Gessel's fundamental quasisymmetric function. The statistic recorded here is the number of $\mathfrak{gl}_2$-descents in the bicoloured Motzkin path corresponding to the Dyck path. Restricting to Motzkin paths without coloured steps one obtains the quasisymmetric expansion for the Frobenius character of $\mathfrak S_r$ acting on $\mathfrak{sl}_2^{\otimes r}$. In this case, the conjecture was shown by Braunsteiner [2].

The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).

The number of distinct diagonal sums of a permutation matrix. For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$$ are $(1,0,1,0,2,0)$, so the statistic is $3$.

The detour number of a graph. This is the number of vertices in a longest induced path in a graph. Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.

The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.

The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.