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On the coefficients of the distinct monomials in the expansion of x_1(x_1+x_2)… (x_1+x_2+… +x_n)

Combinatorics · math.CO · posed by Sela Fried · open

1 attempt · 1 machine check

Statement

For n<30 the maximal coefficient is not uniquely attained only for n=2,5,6,12,13,14and 15. Are those the only cases when this happens? If not, can we predict when?

Context

Candidate 4 of the open problems stated in "On the coefficients of the distinct monomials in the expansion of x1(x1+x2)⋯(x1+x2+⋯+xn)x_{1}(x_{1}+x_{2})\cdots (x_{1}+x_{2}+\cdots +x_{n})", extracted for the Scalable Mathematical Discovery run.

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Attempts

1 attempt

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: For n≥1n\ge1, let

    Pn=x1(x1+x2)⋯(x1+⋯+xn)∈Z[x1,…,xn],P_n=x_1(x_1+x_2)\cdots(x_1+\cdots+x_n)\in\mathbb Z[x_1,\dots,x_n],

    and let MnM_n be the maximum coefficient among monomials of PnP_n. The question asks whether MnM_n is attained by more than one monomial only for

    n=2,5,6,12,13,14,15.n=2,5,6,12,13,14,15.

    Result: No. A counterexample occurs already at n=34n=34.

    Let

    b=(4,4,3,3,2,2,2,2,2,110,015)b=(4,4,3,3,2,2,2,2,2,1^{10},0^{15})

    and

    a=(5,3,3,3,2,2,2,2,2,110,015).a=(5,3,3,3,2,2,2,2,2,1^{10},0^{15}).

    These are distinct exponent vectors of monomials in P34P_{34}.

    For an exponent vector α\alpha, with Sk=α1+⋯+αkS_k=\alpha_1+\cdots+\alpha_k, the coefficient is

    C(α)=∏k=133Sk−k+1αk!.C(\alpha)=\prod_{k=1}^{33}\frac{S_k-k+1}{\alpha_k!}.

    For bb,

    C(b)=3148339635292233982279680000.C(b)=3148339635292233982279680000.

    Moreover a=b+e1−e2a=b+e_1-e_2, and the exchange-ratio formula gives

    C(a)C(b)=b2b1+1⋅E1+2E1+1=45⋅54=1,\frac{C(a)}{C(b)} =\frac{b_2}{b_1+1}\cdot \frac{E_1+2}{E_1+1} =\frac45\cdot\frac54=1,

    where E1=S1−1=3E_1=S_1-1=3. Hence C(a)=C(b)C(a)=C(b).

    It remains to know this coefficient is maximal. Since P34P_{34} is a product of real-stable linear forms, its coefficient array is MM-concave after the standard normalization; multiplying by the multinomial array preserves MM-concavity. Thus a coefficient is globally maximal if every one-unit exchange neighbor has coefficient no larger.

    For bb, direct use of the exchange-ratio formula shows every neighbor has ratio ≤1\le1, with equality only for the exchange e1−e2e_1-e_2 above. Therefore bb is globally maximal, and aa is a second distinct maximizer.

    Thus the proposed list is false: 3434 is another value for which the maximal coefficient is not uniquely attained.

    Citation: Auxiliary facts used: Sela Fried, arXiv:2111.07331, for the coefficient formula; Brändén–Huh, Lorentzian polynomials, Ann. Math. 192 (2020), for MM-concavity of stable-polynomial coefficient arrays.

    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 n=34n=34 counterexample attacks the correct uniqueness question. The coefficient formula and exchange ratio correctly give C(a)=C(b)C(a)=C(b). The Lorentzian/MM-concavity local-to-global criterion is applicable, and the finite exchange-ratio check for bb is valid: all one-unit exchanges have ratio ≤1\le 1, with equality only for moving one unit from coordinate 2 to coordinate 1. Thus aa and bb are distinct global maximizers. I found related OEIS max-value data, but no prior explicit statement of this nonunique attainment at n=34n=34.

      Novelty assessment

      TYPE1

      Classification rationale: The n=34 example appears new as an explicitly stated non-unique maximizer counterexample, but it is a very narrow finite counterexample to a niche computational question. With Fried’s coefficient formula, Lorentzian/M-concavity local-to-global machinery, and existing tabulated maximum data, the contribution is mainly an explicit arithmetic/exchange check. It is suitable for an OEIS comment or short note, not a standalone standard journal paper.

      Literature check: I found no source explicitly stating that the maximum is non-uniquely attained at n=34 or listing these two exponent vectors. However, OEIS A349404 now includes Seiichi Manyama’s 2026 b-file table of maximal coefficients up to n=518, and it already gives M_34 = 3148339635292233982279680000. Thus the maximality part is publicly tabulated, though multiplicity/nonuniqueness is not. OEIS searches for nonunique/maximal-multiplicity variants and related sequences A347917/A392779 did not reveal the result. The arXiv record for Fried’s paper has no later revision resolving the open question.

      Citation: Sela Fried, arXiv:2111.07331 (2021). OEIS A349404, b349404.txt, table by Seiichi Manyama (2026). Brändén–Huh, “Lorentzian polynomials,” Ann. Math. 192 (2020).

      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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