The chromatic discriminant of a graph.
The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$.
According to [1], it equals the cardinality of any of the following sets:
(1) Acyclic orientations of G with unique sink at $q$,
(2) Maximum $G$-parking functions relative to $q$,
(3) Minimal $q$-critical states,
(4) Spanning trees of G without broken circuits,
(5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$,
(6) Multilinear Lyndon heaps on $G$.
In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.

Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Return the Dyck path obtained by adding an initial up and a final down step.

The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.