Asymptotic formulas related to the \(M_2\)-rank of partitions without repeated odd parts

  • Chris Jennings-ShafferEmail author
  • Dillon Reihill


We give asymptotic expansions for the moments of the \(M_2\)-rank generating function and for the \(M_2\)-rank generating function at roots of unity. For this we apply the Hardy–Ramanujan circle method extended to mock modular forms. Our formulas for the \(M_2\)-rank at roots of unity lead to asymptotics for certain combinations of N2(rmn) (the number of partitions without repeated odd parts of n with \(M_2\)-rank congruent to r modulo m). This allows us to deduce inequalities among certain combinations of N2(rmn). In particular, we resolve a few conjectured inequalities of Mao.


Integer partitions Partition ranks Rank differences Asymptotics Circle method Rank inequalities \(M_2\)-rank Harmonic Maass forms Modular forms Mock modular forms Mock theta functions 

Mathematics Subject Classification

Primary 11P82 11P81 



The authors thank Kathrin Bringmann for suggesting this project and for useful comments and discussions.


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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of CologneCologneGermany

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