Abstract
For two integers \(1\le j\le k\), we define (k, j)-colored partitions to be those partitions in which parts may appear in k different types and at most j types can appear for a given part size. Let \(c_{k,j}(n)\) be the number of (k, j)-colored partitions of n. Recently, Keith studied (k, j)-colored partitions and proved the following results: For \(j\in \{2,5,8,9\}\), we have \(c_{9,j}(3n+2)\equiv 0\pmod {27}\) for all \(n\ge 0\). For \(j\in \{3,6\}\), we have \(c_{9,j}(9n+2)\equiv 0\pmod {27}\) for all \(n\ge 0\). In this paper, we determine all a, b, c, j with \((a,b)=1\) and \(1\le j\le 8\) such that \(c_{9,j}(an+b)\equiv c\pmod {27}\) for all nonnegative integers n.
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References
Adiga, C., Dasappa, R.: On overpartition pairs into odd parts modulo powers of 2. Discret. Math. 341, 3141–3147 (2018)
Andrews, G.E.: Singular overpartitions. Int. J. Number Theory 11, 1523–1533 (2015)
Chen, S.-C.: On the number of overpartitions into odd parts. Discret. Math. 325, 32–37 (2014)
Chen, S.-C., Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of Andrews’ singular overpartitions. Int. J. Number Theory 11, 1463–1476 (2015)
Chen, W.Y.C., Sun, L.H., Wang, R.-H., Zhang, L.: Ramanujan-type congruences for overpartitions modulo 5. J. Number Theory 148, 62–72 (2015)
Chern, S., Tang, D.: On certain weighted 7-colored partitions. Ramanujan J. 48, 305–322 (2019)
Fu, S., Tang, D.: On a generalized crank for \(k\)-colored partitions. J. Number Theory 184, 485–497 (2018)
Keith, W.J.: Restricted \(k\)-color partitions. Ramanujan J. 40, 71–92 (2016)
Keith, W.J.: Restricted \(k\)-color partitions, II. Int. J. Number Theory 17, 591–601 (2021)
Kim, S.: General colored partition identities. Ann. Comb. 24, 425–438 (2020)
Lovejoy, J.: Overpartitions and real quadratic fields. J. Number Theory 106, 178–186 (2004)
Ma, W.-X., Chen, Y.-G.: On a problem on restricted \(k\)-colored partitions. Int. J. Number Theory 18, 467–472 (2022)
Ray, C., Barman, R.: Infinite families of congruences for \(k\)-regular overpartitions. Int. J. Number Theory 14, 19–29 (2018)
Tang, D.: Congruences modulo powers of 5 for \(k\)-colored partitions. J. Number Theory 187, 198–214 (2018)
Wang, L.: Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions. Discret. Math. 341, 3370–3384 (2018)
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This work was supported by the National Natural Science Foundation of China, No. 12171243 and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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Ma, WX., Chen, YG. On a problem on restricted k-colored partitions, II. Ramanujan J 63, 1109–1118 (2024). https://doi.org/10.1007/s11139-023-00806-1
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DOI: https://doi.org/10.1007/s11139-023-00806-1