Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions
Abstract
Our combinatorial analysis is motivated by the PDE dynamics ut = uxx + g(u), 0<x<1, of solutions u=u(t,x), tโฅ 0, under Neumann boundary conditions. For dissipative nondegenerate nonlinearities g, the global attractors ๐=๐g of the PDE can then be classified by the orderings of their 2n+1 equilibria v at the boundaries x=0,1. We encode the boundary orders as Hamiltonian Sturm permutations. The name ''Sturm'' refers to nodal properties of PDE solutions u(t,x). ''Hamiltonian'' refers to the second order pendulum ODE for equilibria v(x): 0 = vxx + g(v). We determine the generating function a(z)=โn anzn for the counts an of Hamiltonian Sturm permutations. For nโโ, this provides explicit asymptotics of an. We refine these counts as an=โ brq. Here brq 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)=โr,q brqxryq. This implies asymptotically Gaussian distributions of the probabilities pnr=brq/an 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 g(u,ux) under periodic boundary conditions xโ๐1=โ/2โค, where rotating waves arise.
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BibTeX
@misc{fiedler2026counting,
title = {{Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions}},
author = {Bernold Fiedler and Carlos Rocha},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03611},
eprint = {2610.03611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03611},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions
AU - Bernold Fiedler
AU - Carlos Rocha
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03611
UR - https://arxiv.org/abs/2610.03611
AB - Our combinatorial analysis is motivated by the PDE dynamics u_t = u_xx + g(u), 0<x<1, of solutions u=u(t,x), tโฅ 0, under Neumann boundary conditions. For dissipative nondegenerate nonlinearities g, the global attractors ๐=๐_g of the PDE can then be classified by the orderings of their 2n+1 equilibria v at the boundaries x=0,1. We encode the boundary orders as Hamiltonian Sturm permutations. The name ''Sturm'' refers to nodal properties of PDE solutions u(t,x). ''Hamiltonian'' refers to the second order pendulum ODE for equilibria v(x): 0 = v_xx + g(v). We determine the generating function a(z)=โ_n a_nz^n for the counts a_n of Hamiltonian Sturm permutations. For nโโ, this provides explicit asymptotics of a_n. We refine these counts as a_n=โ 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)=โ_r,q b_rqx^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 g(u,u_x) under periodic boundary conditions xโ๐^1=โ/2โค, where rotating waves arise.
ER -