Abstract
Recently, Beck gave the definitions of two partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\), which denote the total number of parts in the partition of n with rank congruent to r modulo m and the total number of ones in the partition of n with crank congruent to r modulo m, respectively. Beck also posed some conjectures on congruences for NT(r, m, n) which were confirmed by Andrews. After that, Chern discovered some new congruences on NT(r, m, n) and \(M_{\omega }(r,m,n)\). Motivated by their works, several identities on NT(r, m, n) and \(M_{\omega }(r,m,n)\) with \(m=5,7\) were established. In this paper, we confirm a conjecture on an identity on NT(r, 9, n) and \(M_{\omega }(r,3,n)\) due to Mao.
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This work was partially supported by National Natural Science Foundation of China (Grant Numbers 12071331, 11971341 and 11971203).
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Mao, R., Xia, E.X.W. A proof of Mao’s conjecture on an identity of Beck’s partition statistics. Ramanujan J 62, 633–648 (2023). https://doi.org/10.1007/s11139-022-00692-z
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DOI: https://doi.org/10.1007/s11139-022-00692-z