Matching interpretation of the period

Establish whether the denominator of the rational toppling script $A_n^{-1}\mathbf1$ has a direct matching or Pfaffian interpretation extending the even-square argument, and determine whether the all-ones class can be transported to a matching action so that $L(n)$ becomes an orbit length.

Background

For even squares, the paper derives the 2-primary order from a domino–sandpile correspondence and a parity argument. No analogous direct matching or Pfaffian interpretation is obtained for the denominator of An−11A_n^{-1}\mathbf1 in general.

The authors discuss Kasteleyn cokernels and actions on perfect matchings as possible tools, but explain that transporting the distinguished all-ones class would require additional control beyond the cited general results.

References

Does the denominator of $A_n{-1}\mathbf1$ have a direct matching or Pfaffian interpretation extending the even-square argument? Beyond the determinant correspondence, Kuperberg studies the associated cokernels . For the planar bipartite graphs obtained by the Kenyon--Propp--Wilson construction, Taylor constructs a simply transitive action of the Kasteleyn cokernel on perfect matchings . In that setting, a specified group element has a concrete action on matchings. Can the all-ones class be transported to such an action so that $L(n)$ becomes an orbit length?

— The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square  (2609.10625 - Akyar et al., 9 Sep 2026) in Section “Further directions,” paragraph “A matching interpretation of the period”

Can an analogous integral description with fixed vertices explain both the folded all-ones class and the remaining $2$-primary factors?

— The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square  (2609.10625 - Akyar et al., 9 Sep 2026) in Section “Further directions,” paragraph “Reflections and the exceptional prime”