• East Asian mathematical journal
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• v.28 no.1
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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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• v.26 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.17 no.3
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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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• v.8 no.2
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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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• v.23 no.4
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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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• v.37 no.1
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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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• v.63 no.2
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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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• v.37 no.3_4
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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.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.61-70
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• 2007

Combinatorics, often called the 21 st century mathematics, has turned out a very important subject for the present information era. Modern combinatorics has started from some mathematical works, for example, Pascal's triangle and the binomial coefficients, and Euler's problems on the partitions of integers and Konigsberg's bridge problem, and so on. In this paper, we investigate the origin of combinatorics by looking over some interesting ancient combinatorial problems and some important problems which have started various subfields of combinatorics. We also discuss a little on the role of combinatorics in mathematics and mathematics education.