The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. Alternatively, remark that the monomials of the polynomial $\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with $n$ east steps: the exponent of $z_i$ is the number of north steps before the $i$-th east step, see [2]. Thus, this statistic records the coefficients of the monomials. A formula for the coefficient of $z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]: $$c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.$$ This polynomial arises in a partial symmetrization process as follows, see [1]. For $w\in\frak{S}_n$, let $w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$. Furthermore, let $$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$$ Then $\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$.

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 number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

The number of successions of a permutation. A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.

The length of the longest pattern of the form k 1 2...(k-1).

The number of rafts of a permutation. Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.