Abstract
Recently, two partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\) which count 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, were introduced by George Beck. Very recently, Mao proved some identities on NT(r, 5, n) and NT(r, 7, n) which imply some congruences due to Andrews, and Chan, Mao and Osburn. In this paper, we prove some identities on Beck’s partition statistic \(M_{\omega }(r,m,n)\) which imply some congruences modulo 7 for \(M_{\omega }(r,7,n)\). Those congruences were conjectured by Chan, Mao and Osburn and confirmed by Chern.
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The authors are very grateful to the referees for their valuable comments and suggestions. This work was supported by the National Science Foundation of China (no. 12371334), the Natural Science Foundation of Jiangsu Province of China (no. BK20221383) and the Jiangsu postgraduate scientific research innovation plan project (no. KYCX21-3370).
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Xia, E.X.W., Yan, F. & Yao, O.X.M. On the total number of ones of partitions associated to cranks. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 167 (2023). https://doi.org/10.1007/s13398-023-01496-6
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DOI: https://doi.org/10.1007/s13398-023-01496-6