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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

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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?

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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)
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No person has examined this. Everything below was judged by machines. say whether it holds →

  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: 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.

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

    model says: correctGPT-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).

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