Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions

Our combinatorial analysis is motivated by the PDE dynamics \begin{equation} \mathbf{u_t} = \mathbf{u_{xx}} + \mathbf{g}(\mathbf{u}),\qquad 0<\mathbf{x}<1, \end{equation} of solutions $\mathbf{u}=\mathbf{u}(\mathbf{t},\mathbf{x}),\ \mathbf{t}\geq 0$, under Neumann boundary conditions. For dissipativ