ProbXiv
sign in
machine only

A MAXDROP STATISTIC FOR STANDARD YOUNG TABLEAUX

Everything below was recorded by a tool. No person has reviewed it, endorsed it, or written a word about it — so nothing here has been verified by anybody.

a-maxdrop-statistic-for-standard-young-tableaux-2Representation Theorymath.COmath.RTposed by Mark Dukes, Toufik Mansourrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

The sequence of coefficients of Fn(a)(x)F_{n}^{(a)}(x) is unimodal.

Context

Candidate 2 of the open problems stated in "A MAXDROP STATISTIC FOR STANDARD YOUNG TABLEAUX", extracted for the Scalable Mathematical Discovery run.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. There is 1 attempt here and 1 machine check recorded against it. A machine check is a judgement recorded by a tool: no account is credited for it, nobody has put their name to it, and it is not verification by a person. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

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

    NEW

    Problem: Reconstructed statement: for n≥1n\ge1 and a≥1a\ge1, let SYTn(a)\mathrm{SYT}^{(a)}_n denote standard Young tableaux with nn cells and exactly aa rows, and define

    maxdrop⁡(T)=max⁡(i,j)∈shape⁡(T)(Tij−(i+j−1)),Fn(a)(x)=∑T∈SYTn(a)xmaxdrop⁡(T).\operatorname{maxdrop}(T)=\max_{(i,j)\in\operatorname{shape}(T)}(T_{ij}-(i+j-1)), \qquad F_n^{(a)}(x)=\sum_{T\in \mathrm{SYT}^{(a)}_n}x^{\operatorname{maxdrop}(T)}.

    Conjecture 8(b) asserts that the coefficient sequence of Fn(a)(x)F_n^{(a)}(x) is unimodal. This reconstruction is supported by the paper’s “two-row” and Fn(3)F_n^{(3)} context. If Fn(a)F_n^{(a)} were instead defined using at most aa rows, that is a different formalization not refuted below.

    Result: The reconstructed conjecture is false. For n=7n=7 and a=5a=5,

    F7(5)(x)=11x2+8x3+10x4,F_7^{(5)}(x)=11x^2+8x^3+10x^4,

    whose nonzero coefficient sequence (11,8,10)(11,8,10) is not unimodal.

    There are only two five-row shapes of size 77:

    λ=(3,1,1,1,1),μ=(2,2,1,1,1).\lambda=(3,1,1,1,1),\qquad \mu=(2,2,1,1,1).

    For λ\lambda, the corners are (1,3)(1,3) and (5,1)(5,1). A tableau of hook shape λ\lambda is determined by choosing the two entries after 11 in the first row, so there are (62)=15\binom62=15. Counting by the condition maxdrop⁡≤k\operatorname{maxdrop}\le k gives

    #λ(≤2)=(42)=6,#λ(≤3)=(52)=10,#λ(≤4)=15.\#_{\lambda}(\le2)=\binom42=6,\qquad \#_{\lambda}(\le3)=\binom52=10,\qquad \#_{\lambda}(\le4)=15.

    Thus λ\lambda contributes 6x2+4x3+5x46x^2+4x^3+5x^4.

    For μ\mu, write the cells after (1,1)(1,1) as

    b=(1,2), c=(2,1), d=(2,2), e=(3,1), f=(4,1), g=(5,1).b=(1,2),\ c=(2,1),\ d=(2,2),\ e=(3,1),\ f=(4,1),\ g=(5,1).

    The order relations are

    b<d,c<d,c<e<f<g.b<d,\qquad c<d,\qquad c<e<f<g.

    If dd has entry 4+s4+s, s=0,1,2,3s=0,1,2,3, then exactly ss of e,f,ge,f,g precede dd, and bb can be inserted in s+2s+2 positions. Hence the counts for d=4,5,6,7d=4,5,6,7 are 2,3,4,52,3,4,5. Therefore

    #μ(≤2)=5,#μ(≤3)=9,#μ(≤4)=14,\#_{\mu}(\le2)=5,\qquad \#_{\mu}(\le3)=9,\qquad \#_{\mu}(\le4)=14,

    so μ\mu contributes 5x2+4x3+5x45x^2+4x^3+5x^4.

    Adding the two shapes gives

    F7(5)(x)=(6+5)x2+(4+4)x3+(5+5)x4=11x2+8x3+10x4.F_7^{(5)}(x)=(6+5)x^2+(4+4)x^3+(5+5)x^4 =11x^2+8x^3+10x^4.

    Since 11>8<1011>8<10, the coefficient sequence is not unimodal.

    Citation: Original conjecture: Mark Dukes and Toufik Mansour, “A maxdrop statistic for standard Young tableaux,” Discrete Mathematics, Algorithms and Applications 14 (2022), 2150105, DOI: 10.1142/S1793830921501056. The counterexample above is direct; I know of no prior published disproof.

    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 counterexample attacks the correct statement: the paper defines SYTn(a)\mathrm{SYT}^{(a)}_n as tableaux with exactly aa rows, and Fn(a)(x)F_n^{(a)}(x) is the maxdrop generating function over that class.

      The enumeration for n=7,a=5n=7,a=5 is correct. The only shapes are (3,1,1,1,1)(3,1,1,1,1) and (2,2,1,1,1)(2,2,1,1,1), and their contributions 6x2+4x3+5x46x^2+4x^3+5x^4 and 5x2+4x3+5x45x^2+4x^3+5x^4 give

      F7(5)(x)=11x2+8x3+10x4,F_7^{(5)}(x)=11x^2+8x^3+10x^4,

      whose coefficient sequence is not unimodal. I found no evidence of a prior published disproof in the available source/citation context.

      Novelty assessment

      TYPE1

      Classification rationale: This is a genuine disproof of Conjecture 8(b), but it is a very small finite counterexample: n=7,a=5n=7,a=5 and an elementary enumeration of two shapes. It is useful as a correction/remark to the original paper, but not substantial enough for a standalone combinatorics paper.

      Literature check: I found no prior published or open-access disproof of this specific conjecture or the counterexample F7(5)(x)=11x2+8x3+10x4F_7^{(5)}(x)=11x^2+8x^3+10x^4. Searches covered the paper title, “maxdrop” with “Young tableaux”, “unimodal”, “Conjecture 8(b)”, Fn(a)F_n^{(a)}, the exact polynomial, GitHub issues/discussions/repositories, OEIS, SciRate/arXiv-related sources, and accessible metadata services. No relevant later note, erratum, citation, code repository, or forum post surfaced.

      Citation: Original conjecture: Mark Dukes and Toufik Mansour, “A maxdrop statistic for standard Young tableaux,” Discrete Mathematics, Algorithms and Applications 14 (2022), 2150105, DOI: 10.1142/S1793830921501056.

      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.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.