Convexity and log-concavity of the partition function weighted by the parity of the crank

2023/07/05 by Janet J. W. Dong, Dong, Janet J. W., Kathy Q. Ji +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.02013

openalex publication_date 2023/07/05 · openalex created_date 2023/07/07 · openalex updated_date 2026/07/28

Abstract

Let M₀(n) (resp. M₁(n)) denote the number of partitions of n with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for M₀(n)-M₁(n). By utilizing this formula with the explicit bound, we show that M_k(n-1)+M_k(n+1)>2M_k(n) for k=0 or 1 and n≥ 39. This result can be seen as the refinement of the classical result regarding the convexity of the partition function p(n), which counts the number of partitions of n. We also show that M₀(n) (resp. M₁(n)) is log-concave for n≥ 94 and satisfies the higher order Turán inequalities for n≥ 207 with the aid of the upper bound and the lower bound for M₀(n) and M₁(n).

Related

• Log-Concavity of the Partition Function
• Partitions weighted by the parity of the crank
• Log-concavity And The Multiplicative Properties of Restricted Partition Functions
• Strict Log-concavity of k-coloured Partitions
• Turán inequalities for the plane partition function