New Congruences for $2$-Color Partitions and $t$-Core Partitions
DOI:
https://doi.org/10.5269/bspm.70916Resumen
Let $c_N(n)$ denote the number of $2$-color partition of $n$ subject to the restriction that one of the colors appears only in parts that are divisible by $N$. If $t$ is a positive integer, then a partition of a nonnegative integer $n$ is a $t$-core if none of the hook numbers of the associated Ferrers-Young diagram is a multiple of $t$. Let $a_t(n)$ denote the number of $t$-core partitions of $n$. In this paper, we obtain new congruences modulo $3$ for the $2$-color partition function $c_{11}(n)$, $t$-core partition functions $a_5(n)$ and $a_{11}(n)$.
Referencias
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J. Sinick, Ramanujan congruences for a class of eta quotients, \emph{Int. J. Number Theory}, 6(4) (2010), 835-847.
H. C. Chan, Ramanujan's cubic continued fraction and an analog of his ``most beautiful identity", \emph{Int. J. Number Theory}, 6(3) (2010), 673--680.
\bibitem{Chan11}
H. H. Chan and P. C. Toh, New analogues of Ramanujan's partition identities, \emph{J. Number Theory}, 130(9) (2010), 1898--1913.
\bibitem{Chern}
S. Chern, New congruences for $2$-color partitions, \emph{J. Number Theory}, 163 (2016), 474--481.
\bibitem{Garvan}
F. Garvan, D. Kim and D. Stanton, Cranks and $t$-cores, \emph{Invent. Math.}, 101(1) (1990), 1--17.
\bibitem{M. D. H3}
M. D. Hirschhorn, The power of $q$, a personal journey, Developments in Mathematics, v. 49,
Springer, 2017.
\bibitem{Liu}
J. Liu and A. Y. Z. Wang, Arithmetic properties of a restricted bipartition function, \emph{Electron. J. Combin.}, 22(3), Paper 3.8, 11 pp.
\bibitem{Ramanujan}
S. Ramanujan, Congruence properties of partitions, \emph{Proc. London Math. Soc.(2)}, 18 (1920), Records for 13 March 1919.
\bibitem{Sinick}
J. Sinick, Ramanujan congruences for a class of eta quotients, \emph{Int. J. Number Theory}, 6(4) (2010), 835-847.
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Publicado
2025-01-21
Número
Sección
Research Articles
Licencia
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
Cómo citar
S. N. Fathima, Utpal Pore, M. A. Sriraj, & P. Siva Kota Reddy. (2025). New Congruences for $2$-Color Partitions and $t$-Core Partitions. Boletim Da Sociedade Paranaense De Matemática, 43, 1-9. https://doi.org/10.5269/bspm.70916
