# P11710: Counting distinct parenthesized powers of omega

- ID: `P11710`
- Reference: `counting-distinct-parenthesized-powers-of-omega`
- Page: https://theoremdb.org/statements/P11710
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/counting-distinct-parenthesized-powers-of-omega?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(F(n)\) be the number of distinct ordinal values obtained by inserting a full parenthesization into a string of \(n\) copies of \(\omega\) joined by exponentiation. Prove \(F(n)=1-n+\sum_{k=1}^{n}t(k)\) for every \(n\ge 1\), where \(t(k)\) counts unlabeled rooted trees with \(k\) vertices.

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `counting-distinct-parenthesized-powers-of-omega`, 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.