Congruences on the number of restricted m-ary partitions

Hou, Qing-Hu; Jin, Hai-Tao; Mu, Yan-Ping; Zhang, Li

doi:10.1016/j.jnt.2016.05.015

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Similar results were obtained for many types of m-ary partitions including the case of m-ary partitions with no gaps [2,6] and m-ary overpartitions [7].…”

Section: Introductionsupporting

confidence: 77%

High order congruences for M-ary partitions

Żmija

2024

Period Math Hung

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For a sequence $$M=(m_{i})_{i=0}^{\infty }$$ M = ( m i ) i = 0 ∞ of integers such that $$m_{0}=1$$ m 0 = 1 , $$m_{i}\ge 2$$ m i ≥ 2 for $$i\ge 1$$ i ≥ 1 , let $$p_{M}(n)$$ p M ( n ) denote the number of partitions of n into parts of the form $$m_{0}m_{1}\cdots m_{r}$$ m 0 m 1 ⋯ m r . In this paper we show that for every positive integer n the following congruence is true: $$\begin{aligned} p_{M}(m_{1}m_{2}\cdots m_{r}n-1)\equiv 0\ \ \left( \textrm{mod}\ \prod _{t=2}^{r}\mathcal {M}(m_{t},t-1)\right) , \end{aligned}$$ p M ( m 1 m 2 ⋯ m r n - 1 ) ≡ 0 mod ∏ t = 2 r M ( m t , t - 1 ) , where $$\mathcal {M}(m,r):=\frac{m}{\textrm{gcd}\big (m,\textrm{lcm}(1,\ldots ,r)\big )}$$ M ( m , r ) : = m gcd ( m , lcm ( 1 , … , r ) ) . Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for m-ary partitions found by Andrews, Gupta, and Rødseth and Sellers.

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“…Similar results were obtained for many types of m-ary partitions including the case of m-ary partitions with no gaps [2,6] and m-ary overpartitions [7].…”

Section: Introductionsupporting

confidence: 77%

High order congruences for M-ary partitions

Żmija

2024

Period Math Hung

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“…We also consider the cases for the m-ary partitions without gaps, wherein if m i is the largest part, then m k for each 0 ≤ k < i also appears as a part. The related works on such restricted m-ary partitions can be found in [2,4,11]. Moreover, in [5,7,10,15], a general class of non-squashing partitions was introduced and studied, which contains m-ary partitions as a special case.…”

Section: Theorem 11mentioning

confidence: 99%

On the enumeration and congruences for m-ary partitions

Sun

Zhang

2018

Journal of Number Theory

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Abstract. Let m ≥ 2 be a fixed integer. Suppose that n is a positive integer such that m j ≤ n < m j+1 for some integer j ≥ 0. Denote b m (n) the number of m-ary partitions of n, where each part of the partition is a power of m. In this paper, we show that b m (n) can be represented as a j-fold summation by constructing a one-to-one correspondence between the m-ary partitions and a special class of integer sequences relying only on the base m representation of n. It directly reduces to Andrews, Fraenkel and Sellers' characterization of the values b m (mn) modulo m. Moreover, denote c m (n) the number of m-ary partitions of n without gaps, wherein if m i is the largest part, then m k for each 0 ≤ k < i also appears as a part. We also obtain an enumeration formula for c m (n) which leads to an alternative representation for the congruences of c m (mn) modulo m due to Andrews, Fraenkel and Sellers.

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“…The notion of m-ary partitions was generalised by many authors and many directions, see for example [3,5,9,13,16,18,17,24]. For us, the two types of generalisations will be important.…”

Section: Introductionmentioning

confidence: 99%