Identify the combinatorial structures for higher-height Morava stabilizer actions

Identify the combinatorial objects that should replace the labelled ordered rooted trees in generalizations of the height-2 formulas for the action of the Morava stabilizer group on Lubin–Tate deformation rings at heights greater than 2.

Background

The paper derives explicit closed formulas, expressed as sums over labelled ordered rooted trees, for the action of the height-2 full Morava stabilizer group on the Lubin–Tate deformation ring. The author then discusses extending these formulas to higher heights.

Although analogous generalizations are expected to be possible, the appropriate combinatorial indexing structures beyond height 2 have not been identified. This is a concrete unresolved component of extending the paper’s formulas to Morava stabilizer groups at heights greater than 2.

References

We close with a remark on the possibility of proving analogues of Theorems A and B at heights h > 2. We expect that such generalizations to higher heights ought to be possible, although we haven't tried to work out those generalizations ourselves, and we expect some patience and some work will be required. We do not know what kinds of combinatorial objects would take the place of p-labelled trees in height h > 2 generalizations.

— The action of the Morava stabilizer group on the coefficients of Morava E-theory at height 2  (2503.04686 - Salch, 6 Mar 2025) in Section 1.3, concluding remark on higher-height analogues of Theorems A and B