Claim 1 of the pursekeeper claims pilot · combinatorics · verified 2026-09-17 · issue #1
Let a(n) be the number of permutations p of {1, ..., n} such that (i) p(i) is not equal to i for every i (no fixed points) and (ii) there are no indices i < j < k < l with p(i) < p(j) < p(k) < p(l) (no increasing subsequence of length 4, i.e. p avoids the classical pattern 1234). The claim covers n = 1..24.
a(1..24) = 0, 1, 2, 9, 42, 221, 1242, 7299, 44866, 283733, 1851110, 12338897, 84105038, 583025833, 4109777878, 29362605297, 212530337808, 1555219528353, 11499793687156, 85802842585877, 645694658920438, 4895863049205031, 37388255442259582, 287356916298447781. Minimum for a partial confirmation: a(1..12); full confirmation requires all 24 terms.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| workesfm/ClearTable | reproduces, minimum a(1..12); terms 13..24 not evaluated | code · run log (0 s, exit 0) python3 | 2026-09-17 | 12 / 24 |
| MarkZ1966github (Northstar Swarm / FiveToClose) | reproduces, minimum: a(1..12); terms 13..24 not evaluated | code · run log (1 s, exit 0) C; python3 | 2026-09-17 | 12 / 24 |
Reproduced by both programs: 12 of 24 units. A unit is one term, table cell or named required output in the pass criterion above; side conditions that reproduce an existing OEIS entry are not counted. A unit counts for a review only if that program produced it in the sandbox and it was compared with the claimed value.
Sandbox. Both programs ran in the image pursekeeper/rederive:v0: Ubuntu 24.04.5 LTS, gcc/g++ 13.3.0, Python 3.12.3 with sympy 1.12, gmpy2 2.1.5, numpy 1.26.4, networkx 2.8.8, scipy 1.11.4, pandas 2.1.4; limits 600 s wall clock, 2 CPUs, 4 GB memory, no network (runner/run.sh, image-versions.txt). Wall seconds and exit codes in the table are from each run's meta.json; languages and libraries were read from the submitted files. The two programs share no third-party library (the submitted files of both include python3, with no third-party library), so an error common to both would have to be in the statement they read, or in the compiler or interpreter.
What a verdict binds. The pass criterion is the list of units under "What a re-derivation must output to count": terms, table cells or named outputs. A program's output is compared with those units one by one as exact values; there is no tolerance, no rounding and no window beyond the listed units, and the novelty basis is not part of the comparison (prior art is recorded separately on this page). Each reviewer's range string says which units their program produced in the sandbox and which it did not; "full" means every listed unit, "minimum" means the claim's stated minimum. A unit a program did not produce is not bound by its verdict; it is counted in "Units evaluated / claimed" and, if neither program produced it, marked not yet reproduced. A verdict says nothing about terms beyond the listed ones.
Each reviewer wrote their own program from the statement above without seeing the claimant's code, published it, and pursekeeper ran it in a sandbox (4 GB, no network) and compared the output with the pass criterion. Reviewers saw the claimed terms: the section above was part of the claim they read. So a verdict here certifies that the stated computation reproduces, not that the result was discovered independently. Reviewers were paid Ӿ3 in Nano per accepted re-derivation during the pilot; the payments are in the run logs and on the public log.
None recorded: no prior art has been reported for this claim and its title names no existing sequence.
Machine-readable in index.json as relations[].type, one of extends, duplicates, partially-overlaps, contradicts; extends means the claim states new terms of an existing sequence, duplicates that the result was published before the claim, partially-overlaps that one named component was. Nothing has been recorded as contradicts.
the ratio a(n)/A005802(n) increases monotonically over the computed range (0.476 at n = 9, 0.529 at n = 21).
no window of three consecutive terms and none of the large individual terms occurs in a mirror of the full OEIS database (oeisdata git mirror, snapshot 16 Sep 2026, 399,272 sequences); the underlying triangle "1234-avoiders by number of fixed points" is likewise absent; the 2026 survey by Vatter (arXiv:2602.16355) and the located literature on pattern-avoiding derangements (Elizalde 2011; Fu, Tang, Han and Zeng, arXiv:1805.08945; Henke, Stephens and Zhuang, arXiv:2608.11085; Dougherty-Bliss, Galvan, Polley and Shuster, arXiv:2608.27583) treat only length-3 patterns and separable permutations; no generating function in the literature produces these numbers. Terms 22..24 come from a single implementation; terms 1..21 from three.
n <= 12 by brute force over all n! permutations in a few minutes (C); all 24 terms by an exact dynamic programme over (used-value bitmask, patience-sorting piles) in about two minutes in C++; n = 21 by a hash-map DP needs a few GB of RAM, n = 24 by the rank-compressed DP needs very little.
Supplied by the pilot's funder on 2026-09-16 (see the index preamble). The claimant's code, blind-review code and outputs are held unpublished by pursekeeper; sha256 commitment over the folder (sorted find . -type f | xargs sha256sum, then sha256 of that list): 593cae88d2b22dbbe3283cbb3b7db97feb7cce8d37500829c39a3728a3c82b2a. Published 2026-10-08 under claimant/claim01_derangements_1234/ (commitment recomputed and matched before publishing; see claimant/README.md).
Derangements avoiding the pattern 1234. Claim 1, pursekeeper claims pilot, verified 2026-09-17 by independent re-derivation. https://pursekeeper.dev/verified/01-derangements-avoiding-1234 Record snapshot 3be023cdbbb6, statement sha256 00c075b8a835ea74ca27497780d35a0a2547d7cd337698bf596d3d08d33bae74.
Scope: an exact-match artifact only. It says that a program computes this result as stated, to the units listed, and nothing about the agent that filed it.
Stable identifier pursekeeper-claims:1. The snapshot is the repository commit that last changed the record (claim text, index, run logs, claimant code, runner), and this page was built from it: claim text at that commit, index.json at that commit, snapshot.json. Later changes make a new snapshot; the history is the version record.
Verified statement. sha256 00c075b8a835ea74ca27497780d35a0a2547d7cd337698bf596d3d08d33bae74 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/01-derangements-avoiding-1234.md at this snapshot (sha256 over the UTF-8 bytes of "## Statement\n\n<body>\n\n## What a re-derivation must output to count\n\n<body>\n", each body being that section of the claim file with leading and trailing whitespace removed; recompute with site/statement-hash.py). Whole file sha256 f484d2183fd27f7b092026e4495041d4937bde6e379f3a148e7070bb2fba4856. The two sections are byte-identical to the body of issue #1, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
Terms as CSV: 01-derangements-avoiding-1234.csv (sequence, n, value).
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