The clique-coclique number of a graph. This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].

The number of [[/StandardTableaux|standard Young tableaux]] of the partition.

The number of semistandard tableaux on a given integer partition with minimal maximal entry. This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$. Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly, $$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$ where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details.

The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.

The order of promotion on the set of standard tableaux of given shape.

The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. For example, there are eight tableaux of shape $[3,2,1]$ with maximal entry $3$, but two of them have the same weight.

The number of nowhere zero 3-flows of a graph. This is the absolute value of the evaluation of the Tutte polynomial of the graph at $(x,y)=(0,-2)$. For $4$-regular graphs, this coincides with the number of Eulerian orientations.

The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. Put differently, for every vertex $v$ of such a path $P$, there is a vertex $w\in P$ and a vertex $u\not\in P$ such that $(v, u)$ and $(u, w)$ are edges. The length of such a path is $0$ if the graph is a forest. It is maximal, if and only if the graph is obtained from a graph $H$ with a Hamiltonian path by joining a new vertex to each of the vertices of $H$.

The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.

The Ore degree of a graph. This is the maximal Ore degree of an edge, which is the sum of the degrees of its two endpoints.