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A MINIMUM PROBLEM FOR FINITE SETS OF REAL NUMBERS WITH NON-NEGATIVE SUM

Combinatorics · math.CO · posed by G. Chiaselotti, G. Marino, C. Nardi · open

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Statement

(P2) If q is an integer such that γ(n, r) ≤ q ≤ η(n, r), can we find n real numbers a1, …, an, such that r of them are non-negative and the remaining n − r are negative with ∑_{i=1}^n a_i ≥ 0, such that the number of the non-negative sums formed from these numbers is exactly q?

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  • A MINIMUM PROBLEM FOR FINITE SETS OF REAL NUMBERS WITH NON-NEGATIVE SUM
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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: Reconstructed statement: for n≥1n\ge1 and 1≤r≤n1\le r\le n, let N(a)N(a) be the number of nonempty subsets S⊆[n]S\subseteq[n] such that ∑i∈Sai≥0\sum_{i\in S}a_i\ge0, where exactly rr of the aia_i’s are nonnegative and the remaining n−rn-r are negative, with ∑iai≥0\sum_i a_i\ge0. Let γ(n,r)\gamma(n,r) and η(n,r)\eta(n,r) be the minimum and maximum possible values of N(a)N(a). The conjecture asks whether every integer q∈[γ(n,r),η(n,r)]q\in[\gamma(n,r),\eta(n,r)] is attained.

    The intended count is of nonempty subset sums; otherwise the stated maximum 2n−12^n-1 in the source would be impossible. Also, for r=nr=n, the definition gives γ(n,n)=η(n,n)=2n−1\gamma(n,n)=\eta(n,n)=2^n-1, so that case is trivial.

    Result: The conjecture is true.

    For r=nr=n, choose all ai>0a_i>0; every nonempty subset sum is nonnegative, so the only value is 2n−12^n-1.

    Assume 1≤r<n1\le r<n, and put m=n−rm=n-r. It is known, and also follows directly, that

    γ(n,r)=2n−1,η(n,r)=2n−2m.\gamma(n,r)=2^{n-1},\qquad \eta(n,r)=2^n-2^m .

    Choose positive numbers

    p1,…,pr, b1,…,bmp_1,\dots,p_r,\ b_1,\dots,b_m

    with ∑ipi=∑jbj\sum_i p_i=\sum_j b_j, and such that the signed vector

    a0=(p1,…,pr,−b1,…,−bm)a^0=(p_1,\dots,p_r,-b_1,\dots,-b_m)

    has no zero subset sum except the empty set and the full set. Such a choice exists: take p1,…,pr,b1,…,bm−1p_1,\dots,p_r,b_1,\dots,b_{m-1} positive and Q\mathbb Q-linearly independent with ∑pi>∑j<mbj\sum p_i>\sum_{j<m}b_j, then set bm=∑pi−∑j<mbjb_m=\sum p_i-\sum_{j<m}b_j.

    For this initial vector, the total sum is 00. Every proper nonempty subset SS pairs with its complement ScS^c, and their sums are nonzero opposites. Hence exactly one of each complementary pair is nonnegative, while the full set is also nonnegative. Therefore

    N(a0)=2n−22+1=2n−1.N(a^0)=\frac{2^n-2}{2}+1=2^{n-1}.

    Now increase only the positive coordinates:

    ai(t)=pi+tci(1≤i≤r),ar+j(t)=−bj(1≤j≤m),a_i(t)=p_i+t c_i\quad(1\le i\le r),\qquad a_{r+j}(t)=-b_j\quad(1\le j\le m),

    where ci>0c_i>0 are chosen generically. For each subset SS containing at least one positive coordinate, its sum is affine increasing in tt. Purely negative subsets remain negative forever.

    For each initially negative mixed subset SS, there is a unique crossing time

    τS=−∑i∈Sai0∑i∈S∩[r]ci>0.\tau_S=\frac{-\sum_{i\in S}a_i^0}{\sum_{i\in S\cap[r]}c_i}>0.

    Because there are only finitely many such subsets, the cic_i’s may be chosen outside finitely many hyperplanes so that all τS\tau_S are distinct. Then as tt increases, the count N(a(t))N(a(t)) starts at 2n−12^{n-1} and rises by exactly 11 at each crossing.

    For sufficiently large tt, every subset containing at least one positive coordinate has nonnegative sum, and every nonempty purely negative subset remains negative. Thus

    N(a(t))=(2r−1)2m=2n−2m=η(n,r).N(a(t))=(2^r-1)2^m=2^n-2^m=\eta(n,r).

    Since the count increases one at a time from γ(n,r)\gamma(n,r) to η(n,r)\eta(n,r), every integer qq in that interval is attained.

    Audit: the construction keeps exactly rr positive entries and mm negative entries, keeps the total sum nonnegative for all t≥0t\ge0, counts nonempty indexed subset sums, handles r=nr=n separately, and realizes exactly the required qq.

    Citation: Problem source: G. Chiaselotti, G. Marino, C. Nardi, “A minimum problem for finite sets of real numbers with non-negative sum,” arXiv:1102.4761. The proof above gives a complete resolution of P2.

  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 proof attacks the correct P2 statement and is mathematically sound. The zero-total generic initial vector gives exactly 2n−1=γ(n,r)2^{n-1}=\gamma(n,r) nonempty nonnegative subset sums for r<nr<n by complement pairing. Increasing only the positive coordinates preserves the required signs and nonnegative total, makes every mixed subset sum monotone increasing, and—after a generic choice of slopes—causes initially negative mixed subsets to cross zero one at a time. Thus the count realizes every integer up to the final value 2n−2n−r=η(n,r)2^n-2^{n-r}=\eta(n,r). The r=nr=n case is handled separately.

    The genericity and no-extra-zero conditions are justified by the stated Q\mathbb Q-linear independence construction. I found no existing exact resolution of P2 beyond the cited paper’s partial/generalized weighted-Boolean-map result, so this is not merely known.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new, but it is a very short and elementary interpolation argument once the endpoint values γ(n,r)\gamma(n,r) and η(n,r)\eta(n,r) are known. It resolves a narrow question from a relatively obscure paper, but the proof is routine enough that it is unlikely to support a standalone standard combinatorics-journal paper without further extensions.

    Literature check: I found the original arXiv paper and its published version. Both state P2 and give only the weighted-Boolean-map analogue/partial result, not the real-number realization proved here. Searches of arXiv for exact and related phrases (“weighted boolean maps,” “non-negative partial sums,” title phrases) found only the original paper in this topic. DOAJ searches for the title, “nonnegative partial sums,” “positive boolean values,” and related phrases likewise found only the original published paper. I did not find any later paper, note, or open-access source containing this exact interval-realization result or a stronger theorem.

    Citation: G. Chiaselotti, G. Marino, C. Nardi, “A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum,” Journal of Applied Mathematics 2012, Article ID 847958, DOI: 10.1155/2012/847958; arXiv:1102.4761.

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