The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.

Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.

The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.

The tree-depth of a graph. The tree-depth $\operatorname{td}(G)$ of a graph $G$ whose connected components are $G_1,\ldots,G_p$ is recursively defined as $$\operatorname{td}(G)=\begin{cases} 1, & \text{if }|G|=1\\ 1 + \min_{v\in V} \operatorname{td}(G-v), & \text{if } p=1 \text{ and } |G| > 1\\ \max_{i=1}^p \operatorname{td}(G_i), & \text{otherwise} \end{cases}$$ Nešetřil and Ossona de Mendez [2] proved that the tree-depth of a connected graph is equal to its minimum elimination tree height and its centered chromatic number (fewest colors needed for a vertex coloring where every connected induced subgraph has a color that appears exactly once). Tree-depth is strictly greater than [[St000536|pathwidth]]. A [[St001120|longest path]] in $G$ has at least $\operatorname{td}(G)$ vertices [3].

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 order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.