Abstract
Recently, Chern proved a number of congruences modulo 5, 7, 11, and 13 on Beck’s partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\), which enumerate the total number of parts in the partitions of n with rank congruent to r modulo m and the total number of ones in the partitions of n with crank congruent to r modulo m, respectively. In this paper, we prove some identities on NT(r, 5, n) and \(M_{\omega }(r,5,n)\) which are analogous to Ramanujan’s “most beautiful identity.” These identities along with some identities proved by Mao, and Jin, Liu, and Xia imply all congruences modulo 5 on NT(r, 5, n) and \(M_{\omega }(r,5,n)\) discovered by Chern.
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References
Andrews, G.E.: The theory of partitions, Addison-Wesley, Reading (1976); reprinted, Cambridge University Press (1998)
Andrews, G.E.: The Ramanujan-Dyson identities and George Beck’s congruence conjectures. Int. J. Number Theory 17, 239–249 (2021)
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull. Am. Math. Soc. 18, 167–171 (1988)
Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Some properties of partitions. Proc. Lond. Math. Soc. 4, 84–106 (1954)
Chan, S.H., Mao, R., Osburn, R.: Variations of Andrews-Beck type congruences. J. Math. Anal. Appl. 495, 124771 (2021)
Chern, S.: Weighted partition rank and crank moments. I. Andrews–Beck type congruences. In: Proceedings of the conference in honor of Bruce Berndt, accepted
Chern, S.: Weighted partition rank and crank moments II. Odd-order moments. Ramanujan J. 57, 471–485 (2022)
Chern, S.: Weighted partition rank and crank moments III. A list of Andrews-Beck type congruences modulo 5, 7, 11 and 13. Int. J. Number Theory 18, 141–163 (2022)
Dyson, F.J.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)
Fu, S.S., Tang, D.Z.: On a generalized crank for \(k\)-colored partitions. J. Number Theory 184, 485–497 (2018)
Jin, L.X., Liu, E.H., Xia, E.X.W.: Proofs of some conjectures of Chan-Mao-Osburn on Beck’s partition statistics. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116, 135 (2022)
Lin, B.L.S., Peng, L., Toh, P.C.: Weighted generalized crank moments for \(k\)-colored partitions and Andrews-Beck type congruences. Discrete Math. 344, 112450 (2021)
Mao, R.: On total number parts functions associated to ranks of overpartitions. J. Math. Anal. Appl. 506, 125715 (2022)
Mao, R.: On the total number of parts functions associated with ranks of partitions modulo \(5\) and \(7\). Ramanujan J. 58, 1201–1243 (2022)
Mao, R., Xia, E.X.W.: A proof of Mao’s conjecture on an identity of Beck’s partition statistics. Ramanujan J. (2023). https://doi.org/10.1007/s11139-022-00692-z
Ramanujan, S.: Some properties of \(p(n)\), the number of partitons of \(n\). Proc. Camb. Philos. Soc. 19, 214–216 (1919)
Xuan, Y., Yao, O.X.M., Zhou, X.Y.: Andrews-Beck type congruences modulo 2 and 4 for Beck’s partition statistics. Result Math. 78, 1–15 (2023)
Yao, O.X.M.: Proof of a Lin-Peng-Toh’s conjecture on an Andrews-Beck type congruence. Discrete Math. 345, 112672 (2022)
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YC and OXMY wrote the main manuscript text. YC and JJ analyzed the data. JJ and OXMY proved the main results. All authors reviewed the manuscript. All authors contributed equally to the article.
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This work was supported by the Natural Science Foundation of Jiangsu Province of China (BK20221383 and BK20200267), the National Science Foundation of China (12371334) and Qinglan Project.
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Chen, Y., Jin, J. & Yao, O.X.M. Some identities on Beck’s partition statistics. Ramanujan J 63, 699–713 (2024). https://doi.org/10.1007/s11139-023-00781-7
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DOI: https://doi.org/10.1007/s11139-023-00781-7