The size of the largest ordinal summand in the 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 maximal cardinality of a summand in the longest ordinal decomposition of a poset.

The order of promotion on the set of linear extensions of a poset.

The largest part of an integer partition.

The greatest common divisor of the parts of the partition.

The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].

The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.

The 3-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-dynamic chromatic number of a graph.

The logarithm of the number of winning configurations of the lights out game on a graph. In the single player lamps out game, every vertex has two states, on or off. The player can toggle the state of a vertex, in which case all the neighbours of the vertex change state, too. The goal is to reach the configuration with all vertices off.

The number of distinct columns in the nullspace of a graph. Let $A$ be the adjacency matrix of a graph on $n$ vertices, and $K$ a $n\times d$ matrix whose column vectors form a basis of the nullspace of $A$. Then any other matrix $K'$ whose column vectors also form a basis of the nullspace is related to $K$ by $K' = K T$ for some invertible $d\times d$ matrix $T$. Any two rows of $K$ are equal if and only if they are equal in $K'$. The nullspace of a graph is usually written as a $d\times n$ matrix, hence the name of this statistic.

The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.