The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.

The maximal multiplicity of a Laplacian eigenvalue in a graph.

The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.

The number of posets with the same order polynomial. The order polynomial of a poset $P$ is the polynomial $S$ such that $S(m)$ is the number of order-preserving maps from $P$ to $\{1,\dots,m\}$. See sections 3.12 and 3.15 of [1].

The number of posets with the same zeta polynomial. The zeta polynomial $Z$ is the polynomial such that $Z(m)$ is the number of weakly increasing sequences $x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset. See section 3.12 of [1]. Since $$ Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k, $$ where $c_k$ is the number of chains of length $k$, this statistic is the same as the number of posets with the same chain polynomial.

The number of posets with combinatorially isomorphic order polytopes.

The Grundy value for Hackendot on posets. Two players take turns and remove an order filter. The player who is faced with the one element poset looses. This game is a slight variation of Chomp. This statistic is the Grundy value of the poset, that is, the smallest non-negative integer which does not occur as value of a poset obtained by a single move.