Determine the asymptotic growth of all Sturm permutations

Determine the true asymptotic growth of the numbers of all Sturm permutations with 2n+1 equilibria, denoted by a_n^*.

Background

The paper derives asymptotic growth for the substantially smaller class of Hamiltonian Sturm permutations, whose counts are denoted by a_n. It compares these with previously derived counts a_n^* of all Sturm permutations with 2n+1 equilibria and notes that numerical data suggest only the crude heuristic relation a_n^* approximately behaves like a_n squared. The actual asymptotic growth of a_n^* is explicitly stated to be unknown.