{"title":"Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset","authors":"Seung-Il Choi","doi":"10.5666/KMJ.2016.56.3.657","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.657","url":null,"abstract":"Given a partition λ = (λ1, λ2, . . . , λl) of a positive integer n, let Tab(λ, k) be the set of all tabloids of shape λ whose weights range over the set of all k-compositions of n and OPλrev the set of all ordered partitions into k blocks of the multiset {1l2l−1 · · · l1}. In [2], Butler introduced an inversion-like statistic on Tab(λ, k) to show that the rankselected Möbius invariant arising from the subgroup lattice of a finite abelian p-group of type λ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(λ, k) and OP λ̂ . When k = 2, we also introduce a major-like statistic on Tab(λ, 2) and study its connection to the inversion statistic due to Butler. 1. Ordered Partitions of a Multiset Let n be a positive integer. An ordered partition of [n] := {1, 2, . . . , n} is a disjoint union of nonempty subsets of [n], and its nonempty subsets are called blocks. Conventionally we denote by π = B1/B2/ · · · /Bk an ordered partition of [n] into k blocks, where the elements in each block are arranged in the increasing order. The set of all ordered partitions of [n] into k blocks will be denoted by OPkn. In the exactly same manner, one can define an ordered partition of a finite multiset. The set of all ordered partitions of a multiset S will be denoted by OPkS . In particular, in case where S is a multiset given by {1, · · · , 1 } {{ } c1−times , 2, · · · , 2 } {{ } c2−times , · · · · · · , l, · · · , l } {{ } cl−times }, (simply denoted by {1122 · · · ll}), we write OPk(c1,··· ,cl) for OP k S . For each π = B1/B2/ · · · /Bk ∈ OP k S , the type of π is defined by a sequence (b1(π), b2(π), · · · , bk(π)), where bi(π) is the cardinality of Received July 29, 2013; revised March 17, 2014; accepted April 11, 2014. 2010 Mathematics Subject Classification: 05A17, 05A18, 11P81.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"657-668"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Effect of Open Packing upon Vertex Removal","authors":"I. S. Hamid, S. Saravanakumar","doi":"10.5666/KMJ.2016.56.3.745","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.745","url":null,"abstract":". In a graph G = ( V, E ), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. The maximum cardinality of an open packing set is called the open packing number and is denoted by ρ o . In this paper, we examine the effect of ρ o when G is modified by deleting a vertex.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"745-754"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marcel Celaya, Kelly Choo, G. MacGillivray, K. Seyffarth
{"title":"Reconfiguring k-colourings of Complete Bipartite Graphs","authors":"Marcel Celaya, Kelly Choo, G. MacGillivray, K. Seyffarth","doi":"10.5666/KMJ.2016.56.3.647","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.647","url":null,"abstract":"Let H be a graph, and k ≥ χ(H) an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer k0(H) such that H has a cyclic Gray code of its k-colourings for all k ≥ k0(H). For complete bipartite graphs, we prove that k0(K`,r) = 3 when both ` and r are odd, and k0(K`,r) = 4 otherwise.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"647-655"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Invariant of Chen-Kuan for Abelian Varieties","authors":"Hyunsuk Moon","doi":"10.5666/KMJ.2016.56.3.755","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.755","url":null,"abstract":"Let A be an abelian variety over a global field K. We show that, in “many” cases, Chen-Kuan’s invariant M(A[n]), that is the average number of n-torsion points of A over various residue fields of K, has the minimal possible value.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"17 1","pages":"755-761"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"G-vector-valued Sequence Space Frames","authors":"E. Osgooei","doi":"10.5666/KMJ.2016.56.3.793","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.793","url":null,"abstract":"G-vector-valued sequence space frames and g-Banach frames for Banach spaces are introduced and studied in this paper. Also, the concepts of duality mapping and β-dual of a BK-space are used to define frame mapping and synthesis operator of these frames, respectively. Finally, some results regarding the existence of g-vector-valued sequence space frames and g-Banach frames are obtained. In particular, it is proved that if X is a separable Banach space and Y is a Banach space with a Schauder basis, then there exist a Y -valued sequence space Yv and a g-Banach frame for X with respect to Y and Yv.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"793-806"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Note on Cellular Structure of Edge Colored Partition Algebras","authors":"A. J. Kennedy, G. Muniasamy","doi":"10.5666/KMJ.2016.56.3.669","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.669","url":null,"abstract":"In this paper, we study the cellular structure of the G-edge colored partition algebras, when G is a finite group. Further, we classified all the irreducible representations of these algebras using their cellular structure whenever G is a finite cyclic group. Also we prove that the Z/rZ-Edge colored partition algebras are quasi-hereditary over a field of characteristic zero which contains a primitive r root of unity.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"669-682"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniqueness of Entire Functions that Share an Entire Function of Smaller Order with One of Their Linear Differential Polynomials","authors":"Xiao-Min Li, H. Yi","doi":"10.5666/KMJ.2016.56.3.763","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.763","url":null,"abstract":". We prove a uniqueness theorem of entire functions sharing an entire function of smaller order with their linear differential polynomials. The results in this paper improve the corresponding results given by Gundersen-Yang[4], Chang-Zhu[3], and others. Some examples are provided to show that the results in this paper are best possible.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"763-776"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A New Kind of Slant Helix in Lorentzian (n + 2)- Spaces","authors":"Fatma Ateş, I. Gök, F. N. Ekmekci","doi":"10.5666/KMJ.2016.56.3.1003","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.1003","url":null,"abstract":"In this paper, we introduce a new kind of slant helix for null curves called null Wn−slant helix and we give a definition of new harmonic curvature functions of a null curve in terms of Wn in (n + 2)−dimensional Lorentzian space M 1 (for n > 3). Also, we obtain a characterization such as: “The curve α is a null Wn − slant helix ⇔ H ′ n − k1Hn−1 − k2Hn−3 = 0” where Hn, Hn−1 and Hn−3 are harmonic curvature functions and k1, k2 are the Cartan curvature functions of the null curve α.","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"1003-1016"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Generalization of Formal Local Cohomology Modules","authors":"S. Rezaei","doi":"10.5666/KMJ.2016.56.3.737","DOIUrl":"https://doi.org/10.5666/KMJ.2016.56.3.737","url":null,"abstract":". Let a and b be two ideals of a commutative Noetherian ring R , M a (cid:12)nitely generated R -module and i an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the i -th a -formal local cohomology module M with respect to b by F i a ; b ( M ). We show that if F i a ; b ( M ) is artinian, then a (cid:18) √ (0 : F i a ; b ( M )). Also, we show that F dim M a ; b ( M ) is artinian and we determine the set Att R F dim M a ; b ( M ).","PeriodicalId":46188,"journal":{"name":"Kyungpook Mathematical Journal","volume":"56 1","pages":"737-743"},"PeriodicalIF":0.7,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70849016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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