Count Neumann connection graphs and global attractors

Determine the counts of directed Neumann connection graphs up to isomorphism and Neumann global attractors up to C^0 orbit equivalence, including for spatially reversible or Hamiltonian nonlinearities.

Background

The paper counts full lap signatures, equivalently Hamiltonian Sturm permutations, but does not identify distinct connection graphs or global attractors under the relevant equivalence relations. Since different Sturm permutations can produce equivalent attractors, counting permutations does not directly solve the enumeration of directed connection graphs or attractors. The authors explicitly state that these counting questions remain unresolved even in the spatially reversible and Hamiltonian classes.

References

The counting results on full lap signatures in our present paper have neither settled the counting of directed Neumann connection graphs $C$, up to isomorphism, nor the counting of Neumann global attractors $A$, up to $C0$ orbit equivalence -- not even for spatially reversible or Hamiltonian nonlinearities $\mathbf{g}\in\Sturm\supset\Sturm$.

— Counting Hamiltonian Sturm permutations: generating functions and Gaussian distributions  (2610.03611 - Fiedler et al., 2 Oct 2026) in Section Periodic boundary conditions and reversibility