On the number of equivalence classes of bi-partitions arising from the color change
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We introduce a new class of bi-partition function $c_k(n)$, which counts the number of bi-color partitions of $n$ in which the second color only appears at the parts that are multiples of $k$. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo $k$. We show that the generating function for $c_k(n)$ involves the partial theta function and obtain the following congruences: \begin{align*} c_2 (27n+26) &\equiv 0 \pmod{3} \\ \intertext{and} c_3 (4n + 2 ) &\equiv 0 \pmod{2}. \end{align*}
Abstract
We introduce a new class of bi-partition function $c_k(n)$, which counts the number of bi-color partitions of $n$ in which the second color only appears at the parts that are multiples of $k$. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo $k$. We show that the generating function for $c_k(n)$ involves the partial theta function and obtain the following congruences: \begin{align*} c_2 (27n+26) &\equiv 0 \pmod{3} \\ \intertext{and} c_3 (4n + 2 ) &\equiv 0 \pmod{2}. \end{align*}
참고문헌 (11)
- Z. Ahmed, "New congruences modulo 5 for the number of 2-color partitions", J. Number Theory, 157, pp.184~198 (2015)[DOI]
- G. E. Andrews, "The Theory of Partitions" (1998), Cambridge Mathematical Library, Cambridge Univ. Press
- G. E. Andrews, "The minimal excludant in integer partitions", J. Integer Seq, 23(2), pp.11~ (2020)
- B. C. Berndt, "Number Theory in the Spirit of Ramanujan" (2006), Amer. Math. Soc
- Z. Cao, "On Somos’ dissection identities", J. Math. Anal. Appl, 365(2), pp.659~667 (2010)[DOI]
- Z. Cao, "Integer matrix exact covering systems and product identities for theta functions", Int. Math. Res. Not. IMRN, 2011(19), pp.4471~4514 (2011)[DOI]
- H. -C. Chan, "Ramanujan’s cubic continued fraction and an analog of his"most beautiful identity"", Int. J. Number Theory, 6(3), pp.673~680 (2010)[DOI]
- G. Gasper, "Basic Hypergeometric Series, second edition" (2004), Cambridge Univ. Press
- J. Huh, "The number of equivalence classes arising from partition involutions", Int. J. Number Theory, 16(5), pp.925~939 (2020)[DOI]
- M. Merca, "A further look at cubic partitions", Ramanujan J, 59(1), pp.253~277 (2022)[DOI]
- S. O. Warnaar, "Partial theta functions", Ramanujan Encyclopedia, ~