# P3662: Longest rowmotion orbit on order ideals of the five cube

- ID: `P3662`
- Reference: `rowmotion-five-cube-max-orbit`
- Page: https://theoremdb.org/statements/P3662
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: b5a83bd9bdbf7dfdc7134c15b7360f889389e7bc
- Current Markdown: https://api.theoremdb.org/v1/statements/rowmotion-five-cube-max-orbit?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(P=[5]\times[5]\times[5]\) with the coordinatewise order. What is the maximum orbit length of rowmotion on the finite set \(J(P)\) of order ideals?

### Problem setup

- **Remark.** Rowmotion sends an order ideal I to the order ideal generated by the minimal elements of P minus I.
- **Definition.** Here [5] is the five-element chain.

### What counts as a solution

- Give the maximum length, a representative orbit, and a complete orbit census whose lengths sum to the number of ideals.

## Status

Status remains unverified. Rowmotion on products of chains has an extensive literature, and the order-five orbit structure may be known computationally.

## Research packet

### Background and intake notes

The order-four case has two orbit lengths, 11 and 33. The larger order has 267227532 ideals, enough to make repeated uncheckpointed enumeration costly.

- Order ideals correspond to plane partitions in a 5 by 5 by 5 box. Enumerate them as monotone height arrays rather than arbitrary 125-bit subsets.
- The map is a permutation, so each state belongs to a cycle and no transient storage is needed. Record one canonical plane partition and a digest for every orbit length.

- Recorded example: For P=[4]^3, exactly six orbits have length 33 and the other 21150 orbits have length 11.

### Computational notes

- A complete exact enumeration of all 232848 ideals of [4]^3 found 21156 rowmotion orbits and maximum length 33.
- MacMahon's boxed plane-partition product gives |J([5]^3)|=267227532 and |J([6]^3)|=1478619421136; the products were evaluated with exact rational arithmetic.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `rowmotion-five-cube-max-orbit`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.