Metadata
bibkey: gawronulas2022partitions
authors: Filip Gawron and Maciej Ulas
year: 2022
title: Signs behaviour of sums of weighted numbers of partitions
doi: 10.1007/s11139-022-00602-3
url: https://arxiv.org/abs/2202.05221v1
claim: "Conjecture 4.4 asserts eventual period-four signs for k ≤ m − 2 and eventual alternating signs for k ≥ m − 1 for the factorial part sets U_m."
strata_touched:
- D5/S1/Recurrence/Partitions/GawronUlasFactorialSignConjecture
license: citation-only
triage: anchor
Verified locator
DOI: 10.1007/s11139-022-00602-3
Source: https://arxiv.org/abs/2202.05221v1
The DOI identifies the publication in Ramanujan Journal 60 (2023), 571–584. The statements quoted below are from the arXiv v1 text. The journal text has not been verified; equality of its conjecture with v1 is ASSUMED-UNVERIFIED.
Source definitions
Pages 1–2: “Let A ⊂ N+ be given and take n ∈ N. By an A-partition λ = (λ_1, . . . , λ_k) with parts in A, of a non-negative integer n, we mean a representation of n in the form n = λ_1 + . . . + λ_k, where λ_i ∈ A. The representations of n differing only in order of the terms are counted as one.” “c_A(i, n) is the number of A-partitions of n with exactly i parts.”
Page 2: “S_{A,k}(n) = Σ_{i=0}^{n} (−1)^i i^k c_A(i, n)”. The empty partition of zero is included, and exponent-zero powers are one.
Conjecture 4.4
Page 11: “Let m ∈ N≥2 and take U_m = {i! : i = 1, . . . , m}. Then, for each k ∈ {0, . . . , m − 2}, (−1)^⌊(n+1)/2⌋ S_{U_m,k}(n) ≥ 0 for n ≫ 0, i.e., the sequence of signs of the sequence (S_{U_m,k}(n)){n∈N} is eventually periodic of period 4. Moreover, if k ≥ m − 1 then (−1)^n S{U_m,k}(n) ≥ 0 for n ≫ 0.”
The formal proposition records both displayed inequalities with separate thresholds for each m and k. It does not strengthen nonnegative signs to strict positivity or require the least eventual period to equal four.
Scope of the refutation
The repository proof refutes the second assertion at m=3 and k=2 by an unbounded family of even arguments. It does not refute the first assertion. The broader literature search and priority of this refutation remain ASSUMED-UNVERIFIED. No claim about a first published refutation is made.