Sums of angles of star polygons and the Eulerian numbers (Q1768038)

Let \(V_1, V_2,\dots, V_n\) be points in the plane such that a convex star polygon \(P\) is formed when consecutive points are joined by directed line segments. Associated with \(P\) is a permutation \(\sigma\in S_n\) such that \(\sigma(k)\) is the position of \(V_k\) in the counterclockwise ordering, with \(\sigma(1)= 1\) in all cases. Two star polygons are said to be combinatorially equivalent if they are associated with the same permutation, up to rotation. The authors derive a formula for the sum of the angles of \(P\) in terms of \(\sigma\). Let \(n_k\) denote the number of combinatorially inequivalent star polygons with \(n\) vertices and sum of angles equal to \(k\pi\), where \(1\leq k\leq n\). The authors derive an explicit formula for \(n_k\), and also express \(n_k\) in terms of Eulerian numbers.