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Inequalities for the \(M_2\)-rank modulo 12 of partitions without repeated odd parts

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Abstract

Let \(N_2(a, M; n)\) denote the number of partitions of n without repeated odd parts whose \(M_2\)-rank is congruent to a modulo M. Lovejoy, Osburn and Mao have found formulas for \(M_2 \)-rank differences modulo 3, 5, 6, and 10. Recently, Xia and Zhao established generating functions for \(N_2(a, 8; n)\) with \(0 \le a\le 7\). Motivated by their works, we establish generating functions for \(N_2(a, 12; n)\) with \(0\le a \le 11\) by using some identities for Appell–Lerch sums and theta functions. Based on these generating functions, we prove some inequalities for certain linear combinations of \(N_2(a,12;n)\) by utilizing asymptotic formulas of eta quotients due to Chern.

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Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu University, Jiangsu, 212013, Zhenjiang, People’s Republic of China

    Yan Fan

  2. School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, 201620, People’s Republic of China

    Eric H. Liu

  3. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, Jiangsu, People’s Republic of China

    Ernest X. W. Xia

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  1. Yan Fan
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  2. Eric H. Liu
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  3. Ernest X. W. Xia
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Correspondence to Eric H. Liu.

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This work was supported by the 2022 Domestic Visiting Scholar Program for Young backbone teachers in Universities of Shanghai (Shanghai University of International Business and Economics), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX21\(\_\)3370), and the National Natural Science Foundation of China (Grant 11971203). .

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Fan, Y., Liu, E.H. & Xia, E.X.W. Inequalities for the \(M_2\)-rank modulo 12 of partitions without repeated odd parts. Ramanujan J 63, 105–130 (2024). https://doi.org/10.1007/s11139-023-00783-5

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  • DOI: https://doi.org/10.1007/s11139-023-00783-5

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