Lattice structures of higher-rank fold-circuit flats

Determine the number of non-isomorphic lattice structures arising from the \(k\)-fold circuits contained in a \(t\)-fold circuit, equivalently the number of rank-\(t\) matroid flat lattices with a prescribed number of principal-partition parts, for principal partitions having more than twelve parts.

Background

The paper explains that extending its almost-coning theorem beyond deletion of two cone edges requires understanding all lower-order fold circuits contained in a tt-fold circuit. These circuits form a lattice isomorphic to the lattice of flats of a rank-tt matroid.

For a 3-fold circuit with twelve principal-partition parts, more than 28 million non-isomorphic lattice structures are possible. The enumeration is explicitly unresolved for larger numbers of parts, illustrating why the extension to t>2t>2 requires methods beyond those developed in the paper.

References

For example, for a $3$-fold circuit whose principal partition has $\ell = 12$ parts, there are over 28 million non-isomorphic lattice structures one could have, and the number is unknown for $\ell > 12$.

— $k$-fold circuits and coning in rigidity matroids  (2508.18838 - Hewetson et al., 26 Aug 2025) in Section 5.3 (Almost coning), discussion following Example~\ref{ex:coning-3-fold}