bibkey: dawsey2022partitionpolynomial authors: Madeline Locus Dawsey; Tyler Russell; Dannie Urban year: 2022 title: “Derivatives and Integrals of Polynomials Associated with Integer Partitions” doi: 10.48550/arXiv.2108.00943 url: https://cs.uwaterloo.ca/journals/JIS/VOL25/Dawsey/dawsey3.pdf claim: “Definition (1) introduces the partition polynomial; Question 9 asks whether derivative values at 1 distinguish any two unequal partitions.” strata_touched:

• D5/S0/Certificates/DawseyPartitionPolynomialDerivativeQuestionRefutation license: citation-only triage: anchor

Partition polynomials and derivative separation

Madeline Locus Dawsey, Tyler Russell, and Dannie Urban published the article in the Journal of Integer Sequences 25 (2022), Article 22.5.1. Printed page 2 states Definition (1):

To define this new polynomial, let λ = ⟨1^{m_1}, 2^{m_2}, . . . , k^{m_k}⟩ be a partition written in frequency notation. We define the partition polynomial f_λ by f_λ(x) = Σ_{i=1}^{k} m_i x^i. (1)

Printed page 9 states the range convention and Question 9:

For the following question, let lg(λ) denote the largest part of the partition λ, and recall that f_λ^{(d)}(x) = 0 for all d > lg(λ).

Question 9. If λ, λ′ are any two unequal partitions, is it true that f_λ^{(d)}(1) ≠ f_{λ′}^{(d)}(1) for some positive integer d ≤ min{lg(λ), lg(λ′)}?

The printed question quantifies over any two unequal partitions and imposes no common-size or common-length hypothesis. The preceding prose motivates the question by asking about unequal partitions even when they have the same length and size. These are distinct readings: the formal claim and its refutation concern the printed quantifiers only and assert nothing about the same-length reading in either direction.

The journal article gives no resolution of Question 9. The arXiv preprint is 2108.00943v2. The recorded literature check found no citing Semantic Scholar paper, no relevant later paper in the authors’ listed work, and no MathDB entry under the article title, its authors, or partition-polynomial derivatives. These bounded searches establish the cited source status used here; they do not claim that no answer exists outside the searched scope.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2108.00943
• URL: https://cs.uwaterloo.ca/journals/JIS/VOL25/Dawsey/dawsey3.pdf
• Journal location: printed pages 2 and 9, Definition (1) and Question 9.