Some New Congruences For Partitions With Monochromatic Even Parts And Multichromatic Odd Parts
Congruences For Partitions With Monochromatic Even Parts And Multichromatic Odd Parts
Résumé
Let $a(n)$ denote the number of integer partitions of a positive integer $n$ wherein even parts appear in only one color (i.e. monochromatic) while the odd parts may appear in one of three colors (i.e. trichromatic). Hirschhorn and Sellers (2025) introduced a generalised version of $a(n)$ and defined the partition function $a_t(n)$ wherein even parts appear in one colour and odd parts may occur with one of the $t$ colours for any fixed positive integer $t$. They also proved some congruences modulo $7$ for $a_{7j+1}(n)$ for any integer $j\ge 1$. In this paper, we prove some new particular and infinite families of congruences for $a_t(n)$ by using $q$-series identities.
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Références
\bibitem{am} T. Amdeberlan and M. Merca, From crank to congruences, https://doi.org/10.48550/arXiv.2505.19991.
\bibitem{br} B. C. Berndt and R. A. Rankin, {\it Ramanujan: Letters and Commentary}, {\it Amer. Math. Soc.} 1995.
\bibitem{b} B. C. Berndt, {\it Ramanujan's Notebook}, Part $III$, Springer-Verlag, New York, 1991.
\bibitem{cui} S. P. Cui and N. S. S. Gu, Arithmetic properties of $l$-regular partitions, {\it Adv. Appl. Math.} {\bf 51} (2013) 507-523.
\bibitem{hirs2} M. D. Hirschhorn, {\it The Power of $q$, A Personal Journey}, Developments in Mathematics, {\bf 49.} Springer, Cham (2017).
\bibitem{hs1} M. D. Hirschhorn and J. A. Sellers, Arithmetic relations for overpartitions, {\it J. Combin. Math. Combin.} {\bf 53} (2005), 65-73.
\bibitem{hs2} M. D. Hirschhorn and J. A. Sellers, A congruence of modulo $3$ for partitions into distinct non multiples of four, {\it J. Integer Seq.} {\bf 17} (2014), Article 14.9.6.
\bibitem{hs3} M. D. Hirschhorn and J. A. Sellers, A Family of Congruences Modulo $7$ for Partitions with Monochromatic Even Parts and Multi--Colored Odd Parts, https://doi.org/10.48550/arXiv.2507.09752.
\bibitem{r} S. Ramanujan, Congruences properties of partitions, {\it Math. Z.} \textbf{9} (1921), 147-153.
\bibitem{r1} S. Ramanujan and G. H. Hardy, {\it Collected papers}, Chelsea, New York, 1962.
\bibitem{toh} P. C. Toh, Ramanujan type identities and congruences for partition pairs, {\it Discret. Math.} {\bf 312} (2012), 1244-1250.
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