Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions

Abstract

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 dissipative nondegenerate nonlinearities $\mathbf{g}$, the global attractors $\mathcal{A}=\mathcal{A}_\mathbf{g}$ of the PDE can then be classified by the orderings of their $2n+1$ equilibria $\mathbf{v}$ at the boundaries $\mathbf{x}=0,1$. We encode the boundary orders as Hamiltonian Sturm permutations. The name ''Sturm'' refers to nodal properties of PDE solutions $\mathbf{u}(\mathbf{t},\mathbf{x})$. ''Hamiltonian'' refers to the second order pendulum ODE for equilibria $\mathbf{v}(\mathbf{x})$: \begin{equation} 0 = \mathbf{v_{xx}} + \mathbf{g}(\mathbf{v}). \end{equation} We determine the generating function $a(z)=\sum_n a_nz^n$ for the counts $a_n$ of Hamiltonian Sturm permutations. For $n\rightarrow\infty$, this provides explicit asymptotics of $a_n$. We refine these counts as $a_n=\sum b_{rq}$. Here $b_{rq}$ counts Hamiltonian Sturm permutations with $2r+1$ spatially homogeneous equilibria and $2q$ spatially non-homogeneous equilibria, such that $r+q=n$. We also determine the explicit generating function $b(x,y)=\sum_{r,q} b_{rq}x^ry^q$. This implies asymptotically Gaussian distributions of the probabilities $p_{nr}=b_{rq}/a_n$ with $r+q=n$, asymptotically for large $n$. We derive asymptotics for means and variances, with error estimates of order $1/n$. All asymptotics are based on work by Flajolet and Sedgewick. We conclude with numerical illustrations and remarks on nonlinearities $\mathbf{g}(\mathbf{u},\mathbf{u_x})$ under periodic boundary conditions $\mathbf{x}\in\mathbb{S}^1=\mathbb{R}/2\mathbb{Z}$, where rotating waves arise.