• East Asian mathematical journal
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• 제28권1호
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• pp.101-107
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• 2012

For each dimension exceeds 1, determining the number of multi-dimensional partitions of a positive integer is an open question in combinatorial number theory. For n ${\leq}$ 14 and d ${\geq}$ 1 we derive a formula for the function ${\wp}_d(n)$ where ${\wp}_d(n)$ denotes the number of partitions of n arranged on a d-dimensional space. We also give an another definition of the d-dimensional partitions using the union of finite number of divisor sets of integers.

• Korean Journal of Mathematics
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• 제26권2호
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• pp.167-174
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• 2018

By acting the dihedral group $D_k$ on the set of k-tuple multi-partitions, we introduce k-gonal partitions for all positive integers k. We give generating functions for these new partition functions and investigate their arithmetic properties.

• 대한수학회논문집
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• 제17권3호
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• pp.431-437
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• 2002

Representing a positive integer in terms of a sum of smaller numbers with certain conditions has been studied since MacMahon [5] pioneered perfect partitions. The complete partitions is in this category and studied by the second author[6]. In this paper, we study complete partitions with more specified completeness, which we call the double-complete partitions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제8권2호
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• pp.1-5
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• 2004

A multiplicative plane partition is a two-dimensional array of positive integers larger than 1 that are nonincreasing both from left to right and top to bottom and whose multiple is a given number n. For a natural number n, let $f_2(n)$ be the number of multiplicative plane partitions of n. In this paper, we prove $f_2(n)\;{\leq}\;n^2$ and a table of them up to $10^5$ is provided.

• Korean Journal of Mathematics
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• 제23권4호
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• pp.637-642
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• 2015

We study congruence properties of the number of cubic partition pairs weighted by the parity of the crank. If we define such number to be c(n), then $c(5n+4){\equiv}0$ (mod 5) and $c(7n+2){\equiv}0$ (mod 7), for all nonnegative integers n.

• East Asian mathematical journal
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• 제37권1호
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• pp.9-17
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• 2021

For a positive integer n, let μd(n) be the number of multiplicative d-dimensional partitions of ${\prod\limits_{i=1}^{n}}p_i$, where pi denotes the ith prime. The number of multiplicative partitions of a square free number with n prime factors is the Bell number μ1(n) = ��n. By the definition of the function μd(n), it can be seen that for all positive integers n, μ1(n) = Tn(1) = ��n, where Tn(x) is the nth Touchard (or exponential ) polynomial. We show that, for a positive n, μ2(n) = 2nTn(1/2). We also conjecture that for all m, μ3(m) ≤ 3mTm(1/3).

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.155-166
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• 2023

In 2018, Andrews introduced the partition functions ${\mathfrak{EO}}$(n) and ${\bar{\mathfrak{EO}}}$(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function ${\mathfrak{EO}}_e$(n) which counts the number of partitions of n which are enumerated by ${\bar{\mathfrak{EO}}}$(n) together with the partitions enumerated by ${\bar{\mathfrak{EO}}}$(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for ${\bar{\mathfrak{EO}}}$(n) and modulo 4 for ${\mathfrak{EO}}_e$(n). For example, if p ≥ 5 is a prime with Legendre symbol $({\frac{-3}{p}})=-1$, then for all integers n ≥ 0 and α ≥ 0, we have ${\bar{\mathfrak{EO}}}(8{\cdot}p^{2{\alpha}+1}(pn+j)+{\frac{19{\cdot}p^{2{\alpha}+2}-1}{3}}){\equiv}0$ (mod 8); 1 ≤ j ≤ (p - 1).

• Journal of applied mathematics & informatics
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• 제37권3_4호
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• pp.177-182
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• 2019

This study is about the number expressed and the number not expressed in terms of the sum of consecutive natural numbers with a difference of 2k. Since it is difficult to generalize in cases of onsecutive positive integers with a difference of 2k, the table of cases of 4, 6, 8, 10, and 12 was examined to find the normality and to prove the hypothesis through the results. Generalized guesswork through tables was made to establish and prove the hypothesis of the number of possible and impossible numbers that are to all consecutive natural numbers with a difference of 2k. Finally, it was possible to verify the possibility and impossibility of the sum of consecutive cases of 124 and 2010. It is expected to be investigated the sum of consecutive natural numbers with a difference of 2k + 1, as a future research task.

• 한국수학사학회지
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• 제20권4호
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• pp.61-70
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• 2007

인류의 문명은 수학적 관찰과 사고의 결과를 정립하고 삶과 자연에 대한 인식과 인식방법을 깨우쳐가며 시작되었다. 수학은 이집트와 이라크(메소포타미아) 등의 중동 지역의 문명에 논리적 사고를 일깨운 그리스-로마 문명이 합쳐지면서 크게 기하학과 대수학의 흐름을 타고 발전하여 왔다. 수학은 다양한 분야로 분파되기도 하고 다시 합쳐지는 과정을 반복하며 발전을 거듭하면서 결국 현대문명의 기반과 토대를 형성하였다. 서양 문명의 역사는 실로 수학의 역사인 것처럼 인식되기도 한다. 20세기 말, 컴퓨터의 발달과 함께 수학에서도 새로운 분야가 태동하여 큰 발전을 보았는데, 이 분야가 이산수학 또는 조합수학이라는 이름으로 불리는 수학이다. 조합수학은 '21세기의 수학'이라는 별칭을 가질 만큼 활성적인 연구 분야로 자리를 잡아가고 있으며 교육적 차원의 중요성도 부각되고 있다. 본 논문에서는 조합수학의 발생을 엿볼 수 있는 흥미로운 문제들을 훑어보며 조합수학의 유래와 의미를 논하고자 한다.