The number of parking functions supported by a Dyck path. One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.

The number of permutations greater than or equal to the given permutation in (strong) Bruhat order.

The number of parabolic double cosets with minimal element being the given permutation. For $w \in S_n$, this is $$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$ where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$. [1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic.

The size of the conjugacy class of a permutation. Two permutations are conjugate if and only if they have the same cycle type, this statistic is then computed as described in [[St000182]].

The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. Given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$. Consider the expansion $$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$ using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$ is a so called unicellular LLT polynomial, and a symmetric function. Consider the Schur expansion $$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$ By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients. Thus, $G_\Gamma(x;q+1)$ is the Frobenius image of some (graded) $S_n$-module. The total dimension of this $S_n$-module is $$D_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1)f^\lambda$$ where $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$. This statistic is $D_\Gamma$.

The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.

The number of endomorphisms of a graph. An endomorphism of a graph $(V, E)$ is a map $f: V\to V$ such that for any edge $(u,v)\in E$ also $\big(f(u), f(v)\big)\in E$.

The size of the largest orbit of antichains under Panyushev complementation.

The rank of the permutation. This is its position among all permutations of the same size ordered lexicographically. This can be computed using the Lehmer code of a permutation: $$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$ where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.

The product of the heights of the descending steps of a Dyck path. A Dyck path with 2n letters defines a partition inside an [n] x [n] board. This statistic counts the number of placements of n non-attacking rooks on the board. By the Gessel-Viennot theory of orthogonal polynomials this corresponds to the 0-moment of the Hermite polynomials. Summing the values of the statistic over all Dyck paths of fixed size n the number of perfect matchings (2n+1)!! is obtained: up steps are openers, down steps closers and the rooks determine a pairing of openers and closers.