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Enumerative and asymptotic analysis of a moduli space

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enumerative-and-asymptotic-analysis-of-a-moduli-spaceRepresentation Theorymath.COmath.RTposed by Margaret A. Readdyrecorded: open · 1 machine check, unexamined

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Statement

For k a non-negative integer and s, t ≥ k, the coefficient of x^{2s-k}e^{tx} in f_{s+t-1}(x) is given by

[x^{2s-k}e^{tx}] f_{s+t-1}(x) = (-1)^{s} \frac{t^{2s+2t-2k-2}}{2^{s-k}\cdot (s-k)! (t-k)!} \cdot Q_{k}(s,t),

where Q_{k}(s,t) is a polynomial in the variables s and t of degree k.

Context

Candidate 1 of the open problems stated in "Enumerative and asymptotic analysis of a moduli space", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Over a characteristic-zero field, let f0(x)=ex−x−1f_0(x)=e^x-x-1, and define fjf_j as in Readdy’s paper by the integral-operator recursion

    Ij+1fj+1=Ij+2fj+1+Ij+1fj+12∑q=0jIq(fq)Ij−q(fj−q),I(g)=∫0xg(u) du.I^{j+1}f_{j+1}=I^{j+2}f_{j+1}+I^{j+1}f_j+ \frac12\sum_{q=0}^j I^q(f_q)I^{j-q}(f_{j-q}), \quad I(g)=\int_0^x g(u)\,du .

    For an expansion fj=∑rpj,r(x)erxf_j=\sum_r p_{j,r}(x)e^{rx}, write [xmetx]fj=[xm]pj,t(x)[x^m e^{tx}]f_j=[x^m]p_{j,t}(x).

    The conjecture claims that for each k≥0k\ge0 and s,t≥ks,t\ge k,

    [x2s−ketx]fs+t−1(x)=(−1)st2s+2t−2k−22s−k(s−k)!(t−k)!Qk(s,t),[x^{2s-k}e^{tx}]f_{s+t-1}(x) = (-1)^s \frac{t^{2s+2t-2k-2}}{2^{s-k}(s-k)!(t-k)!} Q_k(s,t),

    where Qk(s,t)Q_k(s,t) is a polynomial of degree kk.
    The literal endpoint k=s=t=0k=s=t=0 is undefined because f−1f_{-1} is not defined, but the counterexample below is nondegenerate.

    Result: The conjecture is false already for k=2k=2.

    Using the recursion above, one obtains the relevant terms

    f3=(4x4+1603x3+240x2+⋯ )e2x+⋯ ,f4=(218716x4+70474x3+313474x2+⋯ )e3x+⋯ ,f5=(163843x4+6225929x3+9175043x2+⋯ )e4x+⋯ ,f6=(48828125192x4+458984375144x3+22460937516x2+⋯ )e5x+⋯ .\begin{aligned} f_3&=\bigl(4x^4+\tfrac{160}{3}x^3+240x^2+\cdots\bigr)e^{2x}+\cdots,\\ f_4&=\bigl(\tfrac{2187}{16}x^4+\tfrac{7047}{4}x^3+\tfrac{31347}{4}x^2+\cdots\bigr)e^{3x}+\cdots,\\ f_5&=\bigl(\tfrac{16384}{3}x^4+\tfrac{622592}{9}x^3+\tfrac{917504}{3}x^2+\cdots\bigr)e^{4x}+\cdots,\\ f_6&=\bigl(\tfrac{48828125}{192}x^4+\tfrac{458984375}{144}x^3+\tfrac{224609375}{16}x^2+\cdots\bigr)e^{5x}+\cdots . \end{aligned}

    Thus, if Q2Q_2 existed, then for s=2s=2,

    [x2etx]ft+1=t2t−2(t−2)!Q2(2,t),[x^2e^{tx}]f_{t+1}=\frac{t^{2t-2}}{(t-2)!}Q_2(2,t),

    so

    Q2(2,2)=60,Q2(2,3)=3874,Q2(2,4)=4483,Q2(2,5)=17258.Q_2(2,2)=60,\quad Q_2(2,3)=\frac{387}{4},\quad Q_2(2,4)=\frac{448}{3},\quad Q_2(2,5)=\frac{1725}{8}.

    But the restriction of a degree-≤2\le2 polynomial Q2(s,t)Q_2(s,t) to the line s=2s=2 must be a quadratic polynomial in tt, whose third finite difference is zero. Here

    17258−3⋅4483+3⋅3874−60=−178≠0.\frac{1725}{8}-3\cdot\frac{448}{3} +3\cdot\frac{387}{4}-60 = -\frac{17}{8}\ne0.

    Therefore no polynomial Q2(s,t)Q_2(s,t) of degree 22 can satisfy the conjectured formula. This refutes Conjecture 4.1.

    Verification audit: the counterexample uses k=2k=2 and s,t≥2s,t\ge2, so it is not an endpoint or convention issue; the coefficient notation matches Readdy’s exponential-polynomial expansion; and the contradiction rules out both total degree ≤2\le2 and bidegree ≤2\le2 interpretations.

    Citation: Margaret A. Readdy, “Enumerative and asymptotic analysis of a moduli space,” Advances in Applied Mathematics 47 (2011), 575–588; arXiv:1009.4938.

    Reviews

    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The claimed disproof targets the correct statement. For k=2,s=2k=2,s=2, the conjecture would force Q2(2,t)Q_2(2,t) to be a quadratic polynomial in tt. The displayed coefficients obtained from Readdy’s recursion give

      60,  387/4,  448/3,  1725/860,\;387/4,\;448/3,\;1725/8

      for t=2,3,4,5t=2,3,4,5, whose third finite difference is −17/8≠0-17/8\neq0. Thus no degree-2 polynomial Q2Q_2 can satisfy the conjectured formula. The chosen values are within the stated domain, so this is a valid counterexample.

      Novelty assessment

      TYPE1

      Classification rationale: This appears to be a genuine negative resolution of Readdy’s Conjecture 4.1, but only by a small finite counterexample: k=2k=2, s=2s=2, four computed coefficients, and a third-difference obstruction. It is useful as a correction/erratum-level observation, but it gives no broader replacement theorem or structural explanation. On its own it is too minor for a standard standalone combinatorics paper.

      Literature check: I found no prior published or openly available disproof/proof of this conjecture. Searches for Readdy’s paper, “Conjecture 4.1”, Qk(s,t)Q_k(s,t), the coefficient form [x2s−ketx][x^{2s-k}e^{tx}], and the integral recursion did not reveal any follow-up note, erratum, forum post, or paper resolving the conjecture. The arXiv record has no later correcting version; bibliographic metadata also suggests very little follow-up literature.

      Citation: Margaret A. Readdy, “Enumerative and asymptotic analysis of a moduli space,” Advances in Applied Mathematics 47 (2011), 575–588; arXiv:1009.4938; doi:10.1016/j.aam.2010.12.001.

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