# P100020: Does the distinct-cycle-length probability decrease after n=30?

- ID: `P100020`
- Reference: `dclp-problem-eventual-strict-decrease`
- Page: https://theoremdb.org/statements/P100020
- Record maturity: Retired source record

## Problem

Determine whether q_(n+1) is strictly smaller than q_n for every n at least 30.

## Status

The mathematical status has not passed editorial review.

## Work

### Other known results

- **Theorem 1** (established): The logarithmic derivative of the classical product computes every q_n from earlier coefficients. [3](#reference-3)
- **Computation 1** (reproduced): Exact integer arithmetic proves \(q_n<q_{n-1}\) for every \(31\le n\le5000\); proving the same inequality for every \(n\ge5001\) remains open. [3](#reference-3)
- **Proposition 1** (supported): Published analysis gives q_n = e^(-gamma)(1+1/n)+O(log(n)/n^2) and a full expansion. [1](#reference-1)

### Prior approaches

- **Route 1** (supported): The focused audit found the sequence, its limit, and full asymptotics, with no theorem giving the requested threshold. [2](#reference-2) [3](#reference-3) [1](#reference-1) [4](#reference-4)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python program expands the product with a common factorial denominator and checks every adjacent pair.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `dclp-problem-eventual-strict-decrease`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

Reference review is pending for 1 entry. Each affected row names the fields still awaiting review.

1. <a id="reference-1"></a>Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario, and Nicolas Pouyanne, “A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics”. arXiv:math/0606370 (2006). Philippe Flajolet et al., A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, Electronic Journal of Combinatorics 13 (2006), R103, Proposition 1; D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., 1982, pp. 52-54 https://arxiv.org/abs/math/0606370
   - Also cited at Proposition 1 and the distinct-cycle-length example
   - preprint; reference source; arXiv:math/0606370v1; checked 2026-07-24
   - Source use: citation_only
   - For Eventual decrease for distinct cycle lengths in random permutations: Published analysis gives q_n = e^(-gamma)(1+1/n)+O(log(n)/n^2) and a full expansion.
2. <a id="reference-2"></a>D. Lehmer, “On reciprocally weighted partitions”. Acta Arithmetica 21 (1972), 379-388. DOI 10.4064/aa-21-1-379-388. D. H. Lehmer, Acta Arithmetica 21 (1972), 379-388; Flajolet et al., EJC 13 (2006), R103; A. Knopfmacher and R. Warlimont, Australasian Journal of Combinatorics 13 (1996), 151-162 https://doi.org/10.4064/aa-21-1-379-388
   - Also cited at Acta Arithmetica 21 (1972), 379-388
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - For Eventual decrease for distinct cycle lengths in random permutations: The focused audit found the sequence, its limit, and full asymptotics, with no theorem giving the requested threshold.
3. <a id="reference-3"></a>Generating function and recurrence; D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica 21 (1972), 379-388, Theorem 1. Generating function and recurrence; D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica 21 (1972), 379-388, Theorem 1 https://oeis.org/A007838
   - Also cited at A007838, generating function and bibliography
   - Also cited at Exact computation in dclp-artifact-integer-prefix-certificate, reproduced 2026-07-24
   - reference_database; reference source; checked 2026-07-24
   - Source use: citation_only
   - For Eventual decrease for distinct cycle lengths in random permutations: The logarithmic derivative of the classical product computes every q_n from earlier coefficients.
   - Source named by the research packet.
4. <a id="reference-4"></a>Counting permutations and polynomials with a restricted factorization pattern Section 2, the k=1 distinct-cycle-length case https://ajc.maths.uq.edu.au/pdf/13/ocr-ajc-v13-p151.pdf
   - website; reference source; checked 2026-07-24
   - Reference review pending: relevance note.
   - Source use: citation_only