The Hamming dimension of a graph. Let $H(n, k)$ be the graph whose vertices are the subsets of $\{1,\dots,n\}$, and $(u,v)$ being an edge, for $u\neq v$, if the symmetric difference of $u$ and $v$ has cardinality at most $k$. This statistic is the smallest $n$ such that the graph is an induced subgraph of $H(n, k)$ for some $k$.

The rank of the poset.

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

The number of maximal antichains in a poset.

The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.

The order of rowmotion on the set of order ideals of a poset.

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.