Abstract
A bipartition of $n$ is an ordered pair of partitions $(\lambda,\mu)$ such that the sum of all of the parts equals $n$. In this article, we concentrate on the function $c_5(n)$, which counts the number of bipartitions $(\lambda,\mu)$ of $n$ subject to the restriction that each part of $\mu$ is divisible by $5$. We explicitly establish four Ramanujan type congruences and several infinite families of congruences for $c_5(n)$ modulo $3$.
Highlights
In a series of papers [4, 5, 6], Chan studied the arithmetic properties of the cubic partition function a(n), which is defined by ∞ n=0 a(n)qn =1 (q; q)∞(q2; q2)∞ .the electronic journal of combinatorics 22(3) (2015), #P3.8Throughout the paper, we adopt the following standard q-series notation ∞(a; q)∞ = (1 − aqn−1). n=1In [4], Chan proved thatTheorem 1
A bipartition of n is an ordered pair of partitions (λ, μ) such that the sum of all of the parts equals n
We concentrate on the function c5(n), which counts the number of bipartitions (λ, μ) of n subject to the restriction that each part of μ is divisible by 5
Summary
Asked whether there are any other congruences of the following form a( n + k) ≡ 0 (mod ), where is prime and 0 k < . We investigate the bipartition function c5(n) from an arithmetic point of view in the spirit of Ramanujan’s congruences for the standard partition function p(n). On the other hand, applying (4) with q replaced by q3 yields that (q3; q3)2∞(q15; q15)2∞ ≡ ψ(q3)ψ(q9) − q6ψ(q15)ψ(q45) (mod 3).
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