• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.15-25
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• 2022

In 1963, Graham introduced a problem to find integer partitions such that the reciprocal sum of their parts is 1. Inspired by Graham's work and classical partition identities, we show that there is an integer partition of a sufficiently large integer n such that the reciprocal sum of the parts is 1, while the parts satisfy certain congruence conditions.

• East Asian mathematical journal
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• v.35 no.1
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• pp.9-15
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• 2019

Kaavya introduce an involution on the set of partitions with crank 0 and studied the number of partitions of n which are invariant under Kaavya's involution. If a partition ${\lambda}$ with crank 0 is invariant under her involution, we say ${\lambda}$ is a self-conjugate partition with crank 0. We prove that the number of such partitions of n is equal to the number of partitions with rank 0 which are invariant under the usual partition conjugation. We also study arithmetic properties of such partitions and their q-theoretic implication.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.111-117
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• 2022

Let p_a,b,m(n) be the number of integer partitions of n with more parts congruent to a modulo m than parts congruent to b modulo m. We prove that p_a,b,m(n) ≥ p_b,a,m(n) whenever 1 ≤ a < b ≤ m. We also propose some conjectures concerning series with nonnegative coefficients in their expansions.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1017-1024
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• 2023

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1073-1085
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• 2023

A partition of n is called a t-core partition if none of its hook number is divisible by t. In 2019, Hirschhorn and Sellers [5] obtained a parity result for 3-core partition function a3(n). Motivated by this result, both the authors [8] recently proved that for a non-negative integer α, a3αm(n) is almost always divisible by an arbitrary power of 2 and 3 and at(n) is almost always divisible by an arbitrary power of pji, where j is a fixed positive integer and t = pa11pa22···pamm with primes pi ≥ 5. In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for a2(n) and a13(n) modulo 2 which generalizes some results of Das [2].

• 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.

• The Journal of Industrial Distribution & Business
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• v.9 no.5
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• pp.25-35
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• 2018

Purpose - The purpose of this paper is to partition a last-mile delivery network into zones and to determine locations of last mile delivery centers (LMDCs) in Bangkok, Thailand. Research design, data, and methodology - As online shopping has become popular, parcel companies need to improve their delivery services as fast as possible. A network partition has been applied to evaluate suitable service areas by using METIS algorithm to solve this scenario and a facility location problem is used to address LMDC in a partitioned area. Research design, data, and methodology - Clustering and mixed integer programming algorithms are applied to partition the network and to locate facilities in the network. Results - Network partition improves last mile delivery service. METIS algorithm divided the area into 25 partitions by minimizing the inter-network links. To serve short-haul deliveries, this paper located 96 LMDCs in compact partitioning to satisfy customer demands. Conclusions -The computational results from the case study showed that the proposed two-phase algorithm with network partitioning and facility location can efficiently design a last-mile delivery network. It improves parcel delivery services when sending parcels to customers and reduces the overall delivery time. It is expected that the proposed two-phase approach can help parcel delivery companies minimize investment while providing faster delivery services.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.599-613
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• 2018

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.5 no.5
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• pp.1069-1084
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• 2011

This paper considers an optimal base station clustering problem for designing a mobile (wireless) communication network. For a given network with a set of nodes (base stations), the problem is to optimally partition the set of nodes into subsets (each called a cluster) such that the associated inter-cluster traffic is minimized under certain topological constraints and cluster capacity constraints. In the problem analysis, the problem is formulated as an integer programming problem. The integer programming problem is then transformed into a binary integer programming problem, for which the associated linear programming relaxation is solved in a column generation approach assisted by a branch-and-bound procedure. For the column generation, both a heuristic algorithm and a valid inequality approach are exploited. Various numerical examples are solved to evaluate the effectiveness of the LP (Linear Programming) based branch-and-bound algorithm.

• Journal of KIISE:Software and Applications
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• v.34 no.5
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• pp.431-446
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• 2007

Executable composite Web services are selected by binding a given abstract workflow with the specific Web services that satisfy given QoS requirements. Considering the rapidly increasing number of Web services and their highly dynamic QoS environment, the fast selection of composite services is important. This paper presents a method for quality driven comosite Web services selection based on a workflow partition strategy. The proposed method partitions an abstract workflow into two sub-workflows to decrease the number of candidate services that should be considered. The QoS requirement is also decomposed for each partitioned workflow. Since the decomposition of a QoS requirement is based on heuristics, the selection might fail to find composite Web services. To avoid such a failure, the tightness of a QoS requirement is defined and a workflow is determined to be partitioned according to the tightness. A mixed integer linear programming is utilized for the efficient service selection. Experimental results show that the success rate of partitioning is above 99%. Particularly, the proposed method performs faster and selects composite services whose qualities are not significantly different (less than 5%) from the optimal one.