[#P2448] Eventual unimodality of ternary subset-sum polynomials

Problem. For \(n\ge 1\), define \(P_n(x)=\prod_{k=1}^n(1+x^k+x^{2k})\). Is the coefficient sequence of \(P_n\) unimodal for every \(n\ge 11\)?

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Definitions and notation

1Context

Attempts should record the exact overlap inequality used between the three shifted coefficient sequences. Symmetry alone leaves local valleys near the shift boundaries.

2Definitions

Definition 1 (A finite sequence). A finite sequence is unimodal when it weakly increases up to some index and weakly decreases afterward.

Definition 2 (The coefficients are symmetric because P_n). The coefficients are symmetric because P_n is reciprocal up to its total degree.

3What counts as a solution

Prove coefficient unimodality for every n >= 11, or exhibit the least later failure.

1Resolution

What counts as a solution

Answer (Almkvist's theorem settles every n at least 11). The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.^[1]

Resolution argument

For every positive integer \(n\), \[ P_n(q)=\prod_{k=1}^n(1+q^k+q^{2k}) =\prod_{k=1}^n\frac{1-q^{3k}}{1-q^k}. \] Almkvist studied the more general polynomial \[ f_{n,r}(q)=\prod_{k=1}^n\frac{1-q^{rk}}{1-q^k}. \] His 1989 paper proves the conjectured unimodality for \(3\leq r\leq20\), as well as \(r=100,101\), with the odd-\(r\) range beginning at \(n=11\). Setting \(r=3\) gives the candidate verbatim. Hence the coefficient sequence of \(P_n\) is unimodal for every \(n\geq11\).

Dong and Ji identify the result explicitly in their introduction: their Conjecture 1.1 is the displayed product, and the paragraph after equation (1.5) records Almkvist's proved cases \(3\leq r\leq20\). The candidate is a rediscovery of this classical result.

1Packet records

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Record	Kind	Assessment
The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.^[1] Relevance to this problem For Eventual unimodality of ternary subset-sum polynomials, record tspu-claim-almkvist-r3 (“Almkvist's theorem settles every n at least 11”) records a bound, answer, status fact, or structural consequence. The record states: The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every integer n at least 11 Sources Reference 1 Argument For every positive integer \(n\), \[ P_n(q)=\prod_{k=1}^n(1+q^k+q^{2k}) =\prod_{k=1}^n\frac{1-q^{3k}}{1-q^k}. \] Almkvist studied the more general polynomial \[ f_{n,r}(q)=\prod_{k=1}^n\frac{1-q^{rk}}{1-q^k}. \] His 1989 paper proves the conjectured unimodality for \(3\leq r\leq20\), as well as \(r=100,101\), with the odd-\(r\) range beginning at \(n=11\). Setting \(r=3\) gives the candidate verbatim. Hence the coefficient sequence of \(P_n\) is unimodal for every \(n\geq11\). Dong and Ji identify the result explicitly in their introduction: their Conjecture 1.1 is the displayed product, and the paragraph after equation (1.5) records Almkvist's proved cases \(3\leq r\leq20\). The candidate is a rediscovery of this classical result. Related: Proposition 2 Open full record →
Almkvist's theorem and exact small cases classify every positive order.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 2 Almkvist's theorem and exact small cases classify every positive order.^[1] Relevance to this problem For Eventual unimodality of ternary subset-sum polynomials, record tspu-claim-complete-exceptions (“The complete failure set is 2, 4, 6, 8, 9, and 10”) records a bound, answer, status fact, or structural consequence. The record states: Almkvist's theorem and exact small cases classify every positive order. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope every positive integer n Sources Reference 1 Argument Exact convolution shows that \(P_n\) is unimodal for \(n=1,3,5,7\) and fails for \(n=2,4,6,8,9,10\). The descents on the increasing half occur after exponents 2, 8, 18 and 20, 34, 44, and 54, respectively. The corresponding coefficient drops are \[ 2>1,\quad7>6,\quad34>33,\quad36>35,\quad214>213,\quad549>547,\quad1423>1417. \] At \(n=10\), the eleven central coefficients are \[ 1367,1379,1404,1408,1423,1417,1423,1408,1404,1379,1367. \] At \(n=11\), the central valley has disappeared; the corresponding window is \[ 3608,3657,3684,3715,3723,3735,3723,3715,3684,3657,3608. \] Almkvist's theorem covers every \(n\geq11\). Therefore the six listed orders are all the failures. Open full record →
Standard-library Python checks symmetry, total mass, and every adjacent coefficient through order 300.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Standard-library Python checks symmetry, total mass, and every adjacent coefficient through order 300.^[1] Relevance to this problem For Eventual unimodality of ternary subset-sum polynomials, record tspu-artifact-sweep-300 (“Exact coefficient sweep through n = 300”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks symmetry, total mass, and every adjacent coefficient through order 300. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every polynomial order from 1 through 300 Sources Reference 1 Entry point `join source_lines with newline and run with python3` Runtime CPython 3, standard library only Details The program starts with the coefficient vector `[1]` and applies the exact recurrence \[ a_n(j)=a_{n-1}(j)+a_{n-1}(j-n)+a_{n-1}(j-2n), \] with out-of-range terms set to zero. At every order it checks reciprocity, verifies that the coefficient sum is \(3^n\), and scans every adjacent pair through the center. The failure set through 300 is exactly \(\{2,4,6,8,9,10\}\). At every order from 11 through 300 the only equal adjacent pairs before the center occur at exponents 0 and 2. The complete coefficient vectors at \(n=11,100,300\), serialized as compact JSON, have SHA-256 digests `4e87e46d97a8dba7cb176b5437b873de075b81464036efa1e846ed25c99fd74f`, `9c1d89fd72812e83cbacbfe57bfb9ffe9ea618e52b6e0cca079ea16025d6ec0c`, and `6a588b1ee8f54446ea03a2abe977271047d2c84c30c956bf28f5ed20d50f4177`. Related: Proposition 2 · Proposition 1 Open full record →
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Notes and companion material

Original intake status. SOLVED in the reviewed TheoremDB packet as of 2026-08-01. The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.

Induction by convolving with 1+x^n+x^(2n) is the attractive route.
Sparse convolution does not preserve unimodality without overlap inequalities. The failures at n=8,9,10 occur late enough to defeat an induction that checks only symmetry and a central maximum.
Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.

Recorded example 1. Unimodality fails at n=2,4,6,8,9,10 and holds at n=1,3,5,7 within the small cases.

Computational notes

Coefficients were generated by exact integer convolution for every 1 <= n <= 100. The failure set in this range was exactly {2,4,6,8,9,10}.

How the 3 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemEventual unimodality of ternary subset-sum polynomials

All 3 recorded relations between these records and the problem

2See also

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“Eventual unimodality of ternary subset-sum polynomials.” TheoremDB. P2448. Problem statement; statement text SHA-256 b7259d524e15ab01d324d08dae9b9588f530bca7bf5b5e041395ecde28bd0a80. https://theoremdb.org/statement/?ref=P2448

BibTeX

@misc{theoremdb-problem-b7259d524e15ab01d324d08dae9b9588f530bca7bf5b5e041395ecde28bd0a80,
  title = {{Eventual unimodality of ternary subset-sum polynomials}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b7259d524e15ab01d324d08dae9b9588f530bca7bf5b5e041395ecde28bd0a80},
  url = {https://theoremdb.org/statement/?ref=P2448}
}

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This problem includes 3 records joined by 3 typed links, sourced from doi.org^[1], current as of July 24, 2026.

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1References

Packet source. Gert Almkvist, “Proof of a conjecture about unimodal polynomials”. Journal of Number Theory 32(1) (1989), 43-57. DOI 10.1016/0022-314X(89)90096-6. The r=3 specialization of the all-order unimodality theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at G. Almkvist, Proof of a conjecture about unimodal polynomials, Journal of Number Theory 32 (1989), 43-57, r = 3 specialization; cross-identified in Dong and Ji, Unimodality of partition polynomials related to Borwein's conjecture, Introduction, Conjecture 1.1 and the paragraph after equation (1.5).Also cited at Almkvist's all-n result for n at least 11, combined with the exact finite computation in tspu-artifact-sweep-300.Also cited at Inline CPython 3 source below, executed on 2026-07-24.For Eventual unimodality of ternary subset-sum polynomials, the reviewed source scope is G. Almkvist, Proof of a conjecture about unimodal polynomials, Journal of Number Theory 32 (1989), 43-57, r = 3 specialization; cross-identified in Dong and Ji, Unimodality of partition polynomials related to Borwein's conjecture, Introduction, Conjecture 1.1 and the paragraph after equation (1.5). The packet makes no inference beyond that cited scope.Source named by the research packet.

Original coefficient-unimodality target generated by an agent.

[#P2448] Eventual unimodality of ternary subset-sum polynomials

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