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Log-concavity And The Multiplicative Properties of Restricted Partition Functions

Arindam Roy

TL;DR

This paper establishes a general link between eventual log-concavity and multiplicative abundance: any positive sequence that becomes log-concave after a index $N$, and satisfies a concrete initial-condition bound, is eventually abundant in the sense that $x_nx_m \ge x_{n+m}$ beyond a threshold, with explicit bounds tying $x_n$ to early terms. The main result is proved via a chain of log-concavity-derived inequalities and a backward-construction argument that propagates multiplicative structure, supplemented by an extra condition that guarantees the inequality holds for all sufficiently large indices. The framework is then applied to a broad class of restricted partition functions (e.g., $p^k(n)$, $p_\alpha(n)$, $p_\mathcal{A}(n,k)$, and $b^m(n)$), yielding abundance results and detailing the finite set of equalities or failures in small ranges. The findings unify log-behavior and multiplicative inequalities, showing sufficiency without necessity and offering practical criteria to certify abundance for diverse combinatorial sequences with significant implications for partition theory and related combinatorics.

Abstract

The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq p(n+m)$ for $n\geq m\geq 2$ with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence $\{x_n\}$ satisfying a particular initial condition likewise satisfies the inequality $x_nx_m \geq x_{n+m}$. This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.

Log-concavity And The Multiplicative Properties of Restricted Partition Functions

TL;DR

This paper establishes a general link between eventual log-concavity and multiplicative abundance: any positive sequence that becomes log-concave after a index , and satisfies a concrete initial-condition bound, is eventually abundant in the sense that beyond a threshold, with explicit bounds tying to early terms. The main result is proved via a chain of log-concavity-derived inequalities and a backward-construction argument that propagates multiplicative structure, supplemented by an extra condition that guarantees the inequality holds for all sufficiently large indices. The framework is then applied to a broad class of restricted partition functions (e.g., , , , and ), yielding abundance results and detailing the finite set of equalities or failures in small ranges. The findings unify log-behavior and multiplicative inequalities, showing sufficiency without necessity and offering practical criteria to certify abundance for diverse combinatorial sequences with significant implications for partition theory and related combinatorics.

Abstract

The partition function and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: for , and satisfy the inequality: for with only finitely many instances of equality or failure. This paper proves that this is no coincidence, that any log-concave sequence satisfying a particular initial condition likewise satisfies the inequality . This paper further determines that these conditions are sufficient but not necessary and considers various examples to illuminate the situation.
Paper Structure (13 sections, 6 theorems, 46 equations, 4 tables)

This paper contains 13 sections, 6 theorems, 46 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Some Examples of Theorem \ref{['main_theorem1']}
  3. Sufficient but not Necessary
  4. Other Log-Concave Sequences
  5. Proof of Theorem \ref{['main_theorem1']}
  6. Various Partition Functions
  7. The -th Power Partition Function
  8. Fractional Partition Function
  9. The Multiset Partition Function
  10. Log-polynomial Sequences
  11. Sufficient But Not Necessary Condition for Partition Functions
  12. The -ary Partition Function
  13. Final Thoughts

Key Result

Theorem 1.1

Let $\{x_n\}$ be a sequence of positive real numbers and let $N$ be a non-negative integer such that for all $n > N$, the sequence satisfies $x_n^2 \geq x_{n-1}x_{n+1}$. Then for all $n >N$ and $m\geq 0$, the sequence $\{x_n\}$ satisfies: Moreover, if there exists an integer $k\geq 0$ such that then $x_{n+m} \leq x_nx_m$ for all $n,m \geq N+k$. In addition, $\{x_n\}$ is abundant for all $m,n\geq

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Conjecture 3.2
  • Conjecture 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Conjecture 5.1