The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.

The number of descents of a signed permutation. A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.

The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path.

Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$. This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.

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.