# P11108: Record composition enumeration for down-up alternating permutations

- ID: `P11108`
- Reference: `record-composition-enumeration-alternating-permutations`
- Page: https://theoremdb.org/statements/P11108
- 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/record-composition-enumeration-alternating-permutations?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let $n\ge 1$ and let $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ be a composition of $n$, with partial sums $s_j=\alpha_1+\cdots+\alpha_j$. Let $E_m$ be the $m$th Euler zigzag number. For a down-up alternating permutation $w=a_1a_2\cdots a_{2n}$ satisfying $a_1>a_2<a_3>\cdots>a_{2n-1}>a_{2n}$, form the odd-position subword $\widehat w=a_1a_3\cdots a_{2n-1}$. Its record composition lists the gaps between consecutive positions of left-to-right maxima, ending with the final gap to position $n$. If $N(\alpha)$ counts the permutations whose record composition is $\alpha$, prove that $N(\alpha)=\prod_{j=1}^{\ell}\binom{2s_j-1}{2\alpha_j-1}E_{2\alpha_j-1}$.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.
[Inspect the submitted solution](https://theoremdb.org/research/changes/?proposal_id=tdbrpp1%3A740e161fc87548feb87017df72ab52f9)

A solution has been submitted and awaits independent review. Lean source is attached; signed verification is pending.

## 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 `record-composition-enumeration-alternating-permutations`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## Lean verification

A solution has been submitted and awaits independent review. Lean source is attached; signed verification is pending.

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## References

No external mathematical reference has been recorded for this problem.