Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions

Our combinatorial analysis is motivated by the PDE dynamics \begin{equation} ${\mathbf{u}}_{\mathbf{t}}$ = ${\mathbf{u}}_{\mathbf{x}\mathbf{x}}$ + $\mathbf{g}$($\mathbf{u}$),\qquad 0<$\mathbf{x}$<1, \end{equation} of solutions $\mathbf{u}=\mathbf{u}(\mathbf{t},\mathbf{x}),\mathbf{t}\ge 0$, 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 = ${\mathbf{v}}_{\mathbf{x}\mathbf{x}}$ + $\mathbf{g}$($\mathbf{v}$). \end{equation} We determine the generating function $a(z)={\sum }_n{a}_n{z}^n$ for the counts $a_n$ of Hamiltonian Sturm permutations. For $n\to \mathrm{\infty }$, 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 $b(x,y)={\sum }_{r,q}{b}_{rq}{x}^r{y}^q$. This implies asymptotically Gaussian distributions of the probabilities $p_{nr}=b_{rq}\mathrm{/}{a}_n$ with $r+q=n$, asymptotically for large $n$. We derive asymptotics for means and variances, with error estimates of order $1\mathrm{/}n$. All asymptotics are based on work by Flajolet and Sedgewick. We conclude with numerical illustrations and remarks on nonlinearities $\mathbf{g}(\mathbf{u},{\mathbf{u}}_{\mathbf{x}})$ under periodic boundary conditions $\mathbf{x}\in {\mathbb{S}}^1=\mathbb{R}\mathrm{/}2\mathbb{Z}$, where rotating waves arise.