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CYCLIC SIEVING ON PERMUTATIONS - AN ANALYSIS OF MAPS AND STATISTICS IN THE FINDSTAT DATABASE

Combinatorics · math.CO · posed by Ashleigh Adams, Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch · open

2 comments

Statement

We leave as Conjecture 5.38 that this also holds for n even.

Record

Source
  • CYCLIC SIEVING ON PERMUTATIONS - AN ANALYSIS OF MAPS AND STATISTICS IN THE FINDSTAT DATABASE
  • FAR
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Conjecture 5.38 is reconstructed as follows. For even n≥2n\ge2, let

    I≤3(σ)=∣{(i,j):1≤i<j≤n, j−i≤3, σi>σj}∣I_{\le3}(\sigma)=|\{(i,j):1\le i<j\le n,\ j-i\le3,\ \sigma_i>\sigma_j\}|

    on SnS_n. The conjecture asserts that the statistic generating polynomial

    Fn(q)=∑σ∈SnqI≤3(σ)F_n(q)=\sum_{\sigma\in S_n} q^{I_{\le3}(\sigma)}

    exhibits the CSP at q=−1q=-1 for fixed-point-free involutions such as reverse and complement; equivalently,

    Fn(−1)=0F_n(-1)=0

    for even nn.

    Result: The conjecture is false. For n=30n=30,

    F30(−1)=∑σ∈S30(−1)I≤3(σ)=1400112794472284160≠0.F_{30}(-1)=\sum_{\sigma\in S_{30}}(-1)^{I_{\le3}(\sigma)} =1400112794472284160\ne0.

    Since reverse and complement have no fixed points on S30S_{30}, CSP would require F30(−1)=0F_{30}(-1)=0, contradiction.

    The value above is obtained by the following exact recurrence. Let Dm(a1,…,as)D_m(a_1,\dots,a_s), s=min⁡(3,m)s=\min(3,m), be the signed sum over permutations of length mm whose last ss entries have ranks a1,…,asa_1,\dots,a_s. Start with D0(())=1D_0(())=1. Appending a new last entry of rank r∈{1,…,m+1}r\in\{1,\dots,m+1\} changes the sign by

    (−1)∣{i:ai≥r}∣,(-1)^{|\{i:a_i\ge r\}|},

    because only inversions from the previous three positions to the new position are newly created. The new state is the last three entries of

    (a1+1a1≥r,…,as+1as≥r,r).(a_1+\mathbf1_{a_1\ge r},\dots,a_s+\mathbf1_{a_s\ge r},r).

    Thus the recurrence exactly enumerates Fm(−1)F_m(-1).

    A direct exact implementation:

    from collections import defaultdict
D = {(): 1}
for m in range(30):
    E = defaultdict(int)
    for state, w in D.items():
        for r in range(1, m + 2):
            c = sum(a >= r for a in state)
            new = tuple(a + (a >= r) for a in state) + (r,)
            E[new[-3:]] += (-1 if c % 2 else 1) * w
    D = {s: v for s, v in E.items() if v}
print(sum(D.values()))

    prints

    1400112794472284160

    Citation: No published resolution is used here; this is a new finite counterexample to the reconstructed conjecture.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The claimed counterexample attacks the correct Conjecture 5.38: Statistic 494 is the number of inversions with position distance at most 3, and for reverse/complement (fixed-point-free involutions on S30S_{30}) CSP requires F30(−1)=0F_{30}(-1)=0.

    The recurrence is mathematically sound: when appending a new final entry, only the previous three positions can create new relevant inversions, so tracking the ranks of the last three entries suffices and the sign update is exact. The provided exact integer computation gives F30(−1)=1400112794472284160≠0F_{30}(-1)=1400112794472284160\ne0, which disproves the conjecture. I found no evidence of an existing published resolution.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears to be a genuinely new finite computational counterexample, but it is narrow and routine: it disproves one recent database-generated conjecture by an exact transfer-matrix recurrence and a single value at n=30n=30. This would merit notifying the original authors or perhaps an erratum/short note, but not a standalone standard combinatorics paper.

    Literature check: I found no evidence that this counterexample or the value

    F30(−1)=1400112794472284160F_{30}(-1)=1400112794472284160

    is already in the literature. The original arXiv paper states Conjecture 5.38 for even nn, says it was checked only up to n≤10n\le 10, and the current arXiv record still describes three conjectures. Searches for the exact integer, “Conjecture 5.38” with “cyclic sieving,” “Statistic 494,” “inversions of distance at most 3,” FindStat/St000494, GitHub issues/discussions/repositories, and OEIS did not reveal a prior resolution.

    Citation: Ashleigh Adams, Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch, “Cyclic sieving on permutations — an analysis of maps and statistics in the FindStat database,” arXiv:2402.16251.

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