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Stable multivariate Eulerian polynomials and generalized Stirling permutations

Combinatorics · math.CO · posed by J. Haglund, Mirkó Visontai · open

2 comments

Statement

It would be interesting to see if a certain multivariate generating polynomial of Mahonian–Eulerian statistics, such as ∑{π∈𝔖_n} q^maj(π) ∏{i∈𝒟(π)} x_{π_i}, is stable.

Context

Candidate 2 of the open problems stated in "Stable multivariate Eulerian polynomials and generalized Stirling permutations", extracted for the Scalable Mathematical Discovery run.

Record

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  • Stable multivariate Eulerian polynomials and generalized Stirling permutations
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  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: Let H={z∈C:Im⁡z>0}\mathbb H=\{z\in\mathbb C:\operatorname{Im}z>0\}. For π=π1⋯πn∈Sn\pi=\pi_1\cdots\pi_n\in\mathfrak S_n, let

    Des⁡(π)={i:1≤i<n, πi>πi+1},maj⁡(π)=∑i∈Des⁡(π)i.\operatorname{Des}(\pi)=\{i:1\le i<n,\ \pi_i>\pi_{i+1}\},\qquad \operatorname{maj}(\pi)=\sum_{i\in\operatorname{Des}(\pi)} i .

    The natural substantive reconstruction is: for every nn and every fixed q>0q>0,

    Fn(q;x)=∑π∈Snqmaj⁡(π)∏i∈Des⁡(π)xπiF_n(q;\mathbf x)=\sum_{\pi\in\mathfrak S_n}q^{\operatorname{maj}(\pi)} \prod_{i\in\operatorname{Des}(\pi)}x_{\pi_i}

    is stable in the variables x1,…,xnx_1,\dots,x_n. If qq is instead treated as an additional stability variable, the statement is already false for n=2n=2, since F2=1+qx2F_2=1+qx_2 vanishes at q=x2=iq=x_2=i. The fixed-positive-qq version is the meaningful repair, and it is also false.

    Result: For n=4n=4, direct enumeration gives

    F4(q)=1+qx2+(q+2q2)x3+(q+3q2+3q3)x4+q3x2x3+(q3+q4+q5)x2x4+(q3+4q4+2q5)x3x4+q6x2x3x4.\begin{aligned} F_4(q)=&1+qx_2+(q+2q^2)x_3+(q+3q^2+3q^3)x_4+q^3x_2x_3\\ &+(q^3+q^4+q^5)x_2x_4+(q^3+4q^4+2q^5)x_3x_4+q^6x_2x_3x_4 . \end{aligned}

    At q=3q=3,

    F4(3)=1+3x2+21x3+111x4+27x2x3+351x2x4+837x3x4+729x2x3x4.F_4(3)=1+3x_2+21x_3+111x_4+27x_2x_3+351x_2x_4+837x_3x_4+729x_2x_3x_4 .

    Choose 0<t<2/30<t<2/3, set

    a=−111+i351,b=it,a=\frac{-111+i}{351},\qquad b=it,

    and define

    c=−1+3a+21b+27ab111+351a+837b+729ab.c=-\frac{1+3a+21b+27ab}{111+351a+837b+729ab}.

    Then Im⁡a>0\operatorname{Im}a>0, Im⁡b>0\operatorname{Im}b>0. Writing

    A=1+3a+21b+27ab,B=111+351a+837b+729ab,A=1+3a+21b+27ab,\qquad B=111+351a+837b+729ab,

    one computes

    A=(239−t13)+i(1117+162t13),A=\left(\frac2{39}-\frac t{13}\right) +i\left(\frac1{117}+\frac{162t}{13}\right), B=−27t13+i(1+7884t13).B=-\frac{27t}{13} +i\left(1+\frac{7884t}{13}\right).

    Thus, since 0<t<2/30<t<2/3,

    Im⁡c=Im⁡(−AB)>0.\operatorname{Im}c =\operatorname{Im}\left(-\frac AB\right)>0.

    Finally F4(3;a,b,c)=A+Bc=0F_4(3;a,b,c)=A+Bc=0. Taking also x1=ix_1=i, all variables lie in H\mathbb H, yet F4(3)F_4(3) vanishes. Hence F4(3;x)F_4(3;\mathbf x) is not stable.

    Verification audit: the counterexample uses the standard descent and major-index conventions, has q=3>0q=3>0, n=4n=4, and all xix_i strictly in the upper half-plane. Thus it refutes the intended fixed-qq stability statement, not merely a degenerate boundary case.

    Citation: No prior resolution is used; the counterexample above is direct. The problem originates from Haglund–Visontai, “Stable multivariate Eulerian polynomials and generalized Stirling permutations.”

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

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

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The counterexample is mathematically valid. The listed F4(q)F_4(q) enumeration is correct, and for q=3q=3 the specialization is affine in x4x_4. With x2=ax_2=a, x3=bx_3=b, and x4=c=−A/Bx_4=c=-A/B, one has B≠0B\ne0 and

    Im⁡c=ℜ(A)ℑ(B)−ℑ(A)ℜ(B)∣B∣2>0\operatorname{Im}c=\frac{\Re(A)\Im(B)-\Im(A)\Re(B)}{|B|^2}>0

    because 0<t<2/30<t<2/3 gives ℜA>0,ℑA>0,ℜB<0,ℑB>0\Re A>0,\Im A>0,\Re B<0,\Im B>0. Thus all variables can be put in the upper half-plane while F4(3)=0F_4(3)=0, disproving stability for fixed positive qq. If qq is instead considered a stability variable, the n=2n=2 argument also disproves it. No fatal gap or mismatch with the target statement.

    Novelty assessment

    TYPE1

    Classification rationale: This appears to be a genuinely new negative answer, but the contribution is very small: an explicit n=4, q=3n=4,\ q=3 counterexample obtained by direct enumeration and a short complex specialization. It is useful as a correction/remark to the Haglund–Visontai question, but by itself is unlikely to support a standalone combinatorics journal paper unless combined with a broader classification or theory.

    Literature check: I found no prior source giving this counterexample or a stronger negative resolution. The original arXiv link in the metadata appears erroneous; the actual paper is Haglund–Visontai, European J. Combin. 33 (2012), 477–487. Searches through arXiv/Crossref and related literature for “stable q-analogues,” “qmajq^{maj} stable Eulerian,” “Mahonian-Eulerian stable,” “descent tops major index stable,” and the citing/adjacent papers turned up only related stable refinements or univariate real-rootedness results, not this multivariate qmajq^{maj} descent-top stability failure.

    Citation: J. Haglund and M. Visontai, “Stable multivariate Eulerian polynomials and generalized Stirling permutations,” European J. Combin. 33 (2012), 477–487, DOI: 10.1016/j.ejc.2011.10.007. Related but not resolving this: C. D. Savage and M. Visontai, “The ss-Eulerian polynomials have only real roots,” Trans. Amer. Math. Soc. 367 (2015), 1441–1466.

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