Log-Concavity of the Partition Function

2013/10/29 by DeSalvo, Stephen, Pak, Igor · 8 citations
#05A17 #11N37 #65G99 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1310.7982

Abstract

We prove that the partition function p(n) is log-concave for all n>25. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for p(n).

Cited by

• Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--Zhang
• Asymptotics and inequalities for the distinct partition function
• Rademacher-type exact formula and higher order Turán inequalities for r-colored ℓ-regular partitions
• Strict Log-concavity of k-coloured Partitions
• Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions
• Rectangle partitions generalizing integer partitions
• c_eff from Surgery and Modularity
• Number of orbits of k-subsets of permutations

Related

• Log-concavity And The Multiplicative Properties of Restricted Partition Functions
• Strict Log-concavity of k-coloured Partitions
• Turán inequalities for the plane partition function
• Log-concavity of the overpartition function
• Log-Concavity of Infinite Product Generating Functions