Annals of Combinatorics· 2026Q2
Log-Concavity and the Multiplicative Properties of Restricted Partition Functions
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- Q2SCImago
- 2026year
Short summary
A new mathematical proof demonstrates that log-concavity in a sequence, along with a specific initial condition, guarantees a multiplicative property ($x_n x_m \ge x_{n+m}$), explaining why this occurs in certain restricted partition functions.
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Key points
- Log-concavity ($p(n)^2 \ge p(n-1)p(n+1)$) is proven to imply a multiplicative property ($x_n x_m \ge x_{n+m}$) for sequences under specific initial conditions.
- This mathematical framework explains why certain restricted partition functions exhibit this multiplicative behavior.
- The paper identifies these conditions as sufficient but not necessary for the multiplicative property.
- Examples are provided to illustrate the conditions and their implications.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract The partition function p ( n ) and many of its related restricted partition functions have recently shown independently to satisfy log-concavity: $$p(n)^2 \ge p(n-1)p(n+1)$$ p ( n ) 2 ≥ p ( n - 1 ) p ( n + 1 ) for $$n\ge 26$$ n ≥ 26 , and satisfy the inequality: $$p(n)p(m) \ge p(n+m)$$ p ( n ) p ( m ) ≥ p ( n + m ) for $$n\ge m\ge 2$$ n ≥ m ≥ 2 with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $$\{x_n\}$$ { x n } satisfying a particular initial condition likewise satisfies the inequality $$x_nx_m \ge x_{n+m}$$ x n x m ≥ x n + m . This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.
The authors' abstract, as published at the source. Annals of Combinatorics, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics