Abstract
Let \({\overline{spt}}(n)\) denote the number of smallest parts in the overpartitions of n where the smallest part is not overlined. In recent years, some congruences for \({\overline{spt}}(n)\) were proved. Bringmann, Lovejoy and Osburn found a congruence modulo 3 for \({\overline{spt}}(n)\). Garvan and Jennings-Shaffer presented a characterization of the parity on \({\overline{spt}}( n)\). Motivated by their works, in this paper, we give a characterization of congruences modulo 4 on \({\overline{spt}}( n)\) based on the generating functions of \({\overline{N}}(1,4,n)\) and \({\overline{M}}(1,4,n)\) which denote the number of overpartitions of n whose rank is congruent to 1 mod 4 and the number of overpartitions of n whose first residual crank is congruent to 1 mod 4, respectively.
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The author cordially thanks the anonymous referees for their helpful suggestions.
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This work was supported by the National Natural Science Foundation of China (No. 11971203) and the Natural Science Foundation of Jiangsu Province of China (No. BK20221383).
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Yao, O.X.M. A characterization of congruences modulo 4 on a SPT function of overpartitions. Ramanujan J 60, 795–808 (2023). https://doi.org/10.1007/s11139-022-00606-z
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DOI: https://doi.org/10.1007/s11139-022-00606-z