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MONOTONICITY OF PERMANENTS OF CERTAIN DOUBLY STOCHASTIC MATRICES

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monotonicity-of-permanents-of-certain-doubly-stochastic-matrices-2Probabilitymath.COmath.PRposed by SEOK-ZUN SONGrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

CONJECTURE M(D) : For all A∈∂Ω(Rn),fA(λ)=per(λA+(1−λ)Dn)A \in\partial \Omega(R_{n}),f_{A}(\lambda)=per(\lambda A+(1-\lambda)D_{n}) is nondecreasing in the interval 0≦λ≦10 \leqq\lambda \leqq 1 .

Context

Candidate 2 of the open problems stated in "MONOTONICITY OF PERMANENTS OF CERTAIN DOUBLY STOCHASTIC MATRICES", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
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    GPT-5.5 xhigh

    NEW

    Problem: Reconstructed statement: with Rn=Jn−InR_n=J_n-I_n,

    Ω(Rn)={A≥0:A1=1, AT1=1, aii=0},\Omega(R_n)=\{A\ge0:A\mathbf1=\mathbf1,\ A^T\mathbf1=\mathbf1,\ a_{ii}=0\},

    Dn=(n−1)−1RnD_n=(n-1)^{-1}R_n, and ∂Ω(Rn)\partial\Omega(R_n) the relative boundary of this derangement face, Song’s Conjecture M(D)M(D) asserts that for every A∈∂Ω(Rn)A\in\partial\Omega(R_n),

    fA(λ)=per⁡(λA+(1−λ)Dn)f_A(\lambda)=\operatorname{per}(\lambda A+(1-\lambda)D_n)

    is nondecreasing on 0≤λ≤10\le\lambda\le1. This reconstruction is supported by the notation Ω(Rn)\Omega(R_n) and the surrounding “face of derangement matrices” context.

    Result: The conjecture is false already for n=4n=4. Let

    A=(08/157/1508/15007/1507/1508/157/1508/150).A=\begin{pmatrix} 0&8/15&7/15&0\\ 8/15&0&0&7/15\\ 0&7/15&0&8/15\\ 7/15&0&8/15&0 \end{pmatrix}.

    Then A∈Ω(R4)A\in\Omega(R_4), and since it has off-diagonal zero entries, A∈∂Ω(R4)A\in\partial\Omega(R_4). With

    D4=13(R4),D_4=\frac13(R_4),

    direct expansion of the permanent gives

    fA(λ)=per⁡(λA+(1−λ)D4)=562λ4−2940λ3+3250λ2+562550625.f_A(\lambda) =\operatorname{per}(\lambda A+(1-\lambda)D_4) =\frac{562\lambda^4-2940\lambda^3+3250\lambda^2+5625}{50625}.

    Hence

    fA′(λ)=4λ(562λ2−2205λ+1625)50625,f_A'(\lambda)=\frac{4\lambda(562\lambda^2-2205\lambda+1625)}{50625},

    so

    fA′(1)=4(562−2205+1625)50625=−85625<0.f_A'(1)=\frac{4(562-2205+1625)}{50625}=-\frac8{5625}<0.

    Thus fAf_A decreases near λ=1\lambda=1. Explicitly,

    fA(31/32)−fA(1)=3514155776>0,f_A(31/32)-f_A(1)=\frac{35}{14155776}>0,

    although 31/32<131/32<1, contradicting nondecreasingness.

    Audit: no extra hypotheses were added; AA is a genuine boundary point of the derangement face, and the conclusion directly contradicts Song’s stated monotonicity.

    Citation: No resolving literature citation is used; the counterexample above disproves Conjecture M(D)M(D) as reconstructed from Song’s statement.

    Reviews

    0 human reviews · 1 machine check

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    • 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 counterexample is valid. The displayed AA is doubly stochastic with zero diagonal, hence A∈Ω(R4)A\in\Omega(R_4), and its additional off-diagonal zero entries put it in the relative boundary ∂Ω(R4)\partial\Omega(R_4).

      The permanent expansion for λA+(1−λ)D4\lambda A+(1-\lambda)D_4 checks out:

      fA(λ)=562λ4−2940λ3+3250λ2+562550625.f_A(\lambda)=\frac{562\lambda^4-2940\lambda^3+3250\lambda^2+5625}{50625}.

      Thus

      fA′(1)=−85625<0,f_A'(1)=-\frac{8}{5625}<0,

      so fAf_A is decreasing as λ\lambda approaches 11 from the left. Equivalently fA(31/32)>fA(1)f_A(31/32)>f_A(1), contradicting nondecreasingness. This directly disproves the stated conjecture.

      Novelty assessment

      TYPE1

      Classification rationale: The counterexample appears genuinely new, but it is a very small, explicit 4×44\times4 computation. It disproves Song’s stated monotonicity conjecture, yet the construction is an elementary one-matrix calculation with no broader structural theorem or method. On its own this would be more like a short note/addendum than a standalone standard journal paper.

      Literature check: I found no prior source resolving Song’s specific Conjecture M(D)M(D), nor any occurrence of the exact derangement-face statement involving Ω(Rn)\Omega(R_n), Dn=Rn/(n−1)D_n=R_n/(n-1), and fA(λ)=per⁡(λA+(1−λ)Dn)f_A(\lambda)=\operatorname{per}(\lambda A+(1-\lambda)D_n). Searches also checked related permanent-monotonicity literature around London/Lih–Wang/Hwang and later minimum-permanent work on faces of the Birkhoff polytope. Those sources concern the classical Birkhoff-polytope monotonicity/minimum-permanent problems, not this particular DnD_n-centered derangement-face monotonicity statement. No source with the displayed n=4n=4 counterexample or an equivalent stronger disproof was located.

      Citation: No prior resolving citation found. Original conjecture: S.-Z. Song, “Monotonicity of permanents of certain doubly stochastic matrices,” KoreaScience record JAKO198811920126315.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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