{"title":"A combinatorial problem and numerical semigroups","authors":"Aureliano M. Robles Pérez, José Carlos Rosales","doi":"10.26493/1855-3974.989.d15","DOIUrl":null,"url":null,"abstract":"Let a = ( a 1 , …, a n ) and b = ( b 1 , …, b n ) be two n -tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: The cardinality of C is equal to g ; If x , y ∈ ℕ \\ {0} and x + y ∈ C , then C ∩ { x , y } ≠ ∅ ; If x ∈ C and ( x − b i ) / a i ∈ ℕ \\ {0} for some i ∈ {1, …, n } , then ( x − b i ) / a i ∈ C ; X ∩ C = ∅ .","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.26493/1855-3974.989.d15","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
Let a = ( a 1 , …, a n ) and b = ( b 1 , …, b n ) be two n -tuples of positive integers, let X be a set of positive integers, and let g be a positive integer. In this work we show an algorithmic process in order to compute all the sets C of positive integers that fulfill the following conditions: The cardinality of C is equal to g ; If x , y ∈ ℕ \ {0} and x + y ∈ C , then C ∩ { x , y } ≠ ∅ ; If x ∈ C and ( x − b i ) / a i ∈ ℕ \ {0} for some i ∈ {1, …, n } , then ( x − b i ) / a i ∈ C ; X ∩ C = ∅ .