The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

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.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The difference of lower and upper interactions. An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.

The descent variation of a composition. Defined in [1].

The Frankl number of a lattice. For a lattice $L$ on at least two elements, this is $$\max_x(|L|-2|[x, 1]|),$$ where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.