The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.

The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

The number of ordinal summands of a poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the length of the longest ordinal decomposition of a poset.

The number of maximal chains of minimal length in a poset.

The number of greedy linear extensions of a poset. A linear extension of a poset $P$ with elements $\{x_1,\dots,x_n\}$ is greedy, if it can be obtained by the following algorithm: * Step 1. Choose a minimal element $x_1$. * Step 2. Suppose $X=\{x_1,\dots,x_i\}$ have been chosen. If there is at least one minimal element of $P\setminus X$ which is greater than $x_i$ then choose $x_{i+1}$ to be any such minimal element; otherwise, choose $x_{i+1}$ to be any minimal element of $P\setminus X$. This statistic records the number of greedy linear extensions.