We explore both automated and human approaches to the generalized Missionaries and Cannibals problem.
我们探索了广义传教士和食人族问题的自动化和人工方法。
{"title":"Variations on the Missionaries and Cannibals Problem","authors":"G. Spahn, D. Zeilberger","doi":"10.47443/dml.2022.186","DOIUrl":"https://doi.org/10.47443/dml.2022.186","url":null,"abstract":"We explore both automated and human approaches to the generalized Missionaries and Cannibals problem.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42987171","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}
Let f : A → B be a ring homomorphism of the commutative rings A and B with identities. Let J be an ideal of B . The amalgamation of A with B along J with respect to f is a subring of A × B given by A (cid:46)(cid:47) f J := { ( a, f ( a )+ j ) | a ∈ A , j ∈ J } . In this paper, we investigate the comaximal ideal graph and the comaximal graph of the amalgamated algebra A (cid:46)(cid:47) f J . In particular, we determine the Jacobson radical of A (cid:46)(cid:47) f J , characterize the diameter of the comaximal ideal graph of A (cid:46)(cid:47) f J , and investigate the clique number as well as the chromatic number of this graph.
{"title":"On the Comaximal (Ideal) Graph Associated With Amalgamated Algebra","authors":"Zinat Rastgar, K. Khashyarmanesh, M. Afkhami","doi":"10.47443/dml.2022.095","DOIUrl":"https://doi.org/10.47443/dml.2022.095","url":null,"abstract":"Let f : A → B be a ring homomorphism of the commutative rings A and B with identities. Let J be an ideal of B . The amalgamation of A with B along J with respect to f is a subring of A × B given by A (cid:46)(cid:47) f J := { ( a, f ( a )+ j ) | a ∈ A , j ∈ J } . In this paper, we investigate the comaximal ideal graph and the comaximal graph of the amalgamated algebra A (cid:46)(cid:47) f J . In particular, we determine the Jacobson radical of A (cid:46)(cid:47) f J , characterize the diameter of the comaximal ideal graph of A (cid:46)(cid:47) f J , and investigate the clique number as well as the chromatic number of this graph.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45130722","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}
A set S of vertices in a connected graph G is an irregular dominating set if the vertices of S can be labeled with distinct positive integers in such a way that for every vertex v of G , there is a vertex u ∈ S such that the distance from u to v is the label assigned to u . If for every vertex u ∈ S , there is a vertex v of G such that u is the only vertex of S whose distance to v is the label of u , then S is a minimal irregular dominating set. A graph H is an irregular domination graph if there exists a graph G with a minimal irregular dominating set S such that H is isomorphic to the subgraph G [ S ] of G induced by S . In this paper, all irregular domination trees and forests are characterized. All disconnected irregular domination graphs are determined as well.
{"title":"Irregular Domination Trees and Forests","authors":"Caryn Mays, Ping Zhang","doi":"10.47443/dml.2022.119","DOIUrl":"https://doi.org/10.47443/dml.2022.119","url":null,"abstract":"A set S of vertices in a connected graph G is an irregular dominating set if the vertices of S can be labeled with distinct positive integers in such a way that for every vertex v of G , there is a vertex u ∈ S such that the distance from u to v is the label assigned to u . If for every vertex u ∈ S , there is a vertex v of G such that u is the only vertex of S whose distance to v is the label of u , then S is a minimal irregular dominating set. A graph H is an irregular domination graph if there exists a graph G with a minimal irregular dominating set S such that H is isomorphic to the subgraph G [ S ] of G induced by S . In this paper, all irregular domination trees and forests are characterized. All disconnected irregular domination graphs are determined as well.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43347341","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}
Nuttanon Songsuwan, Supida Sengsamak, Nutchapol Jeerawattana, T. Jiarasuksakun, P. Kaemawichanurat
For the positive integers r and n satisfying r ≤ n , let P r,n be the family of partial permutations {{ (1 , x 1 ) , (2 , x 2 ) , . . . , ( r, x r ) } : x 1 , x 2 , . . . , x r are different elements of { 1 , 2 , . . . , n }} . The subfamilies A 1 , A 2 , . . . , A k of P r,n are called cross intersecting if A ∩ B (cid:54) = ∅ for all A ∈ A i and B ∈ A j , where 1 ≤ i (cid:54) = j ≤ k . Also, if A 1 , A 2 , . . . , A k are mutually disjoint, then they are called disjoint cross intersecting subfamilies of P r,n . For the disjoint cross intersecting subfamilies A 1 , A 2 , . . . , A k of P n,n , it follows from the AM-GM inequality that (cid:81) ki =1 |A i | ≤ ( n ! /k ) k . In this paper, we present two proofs of the following statement: (cid:81) ki =1 |A i | = ( n ! /k ) k if and only if n = 3 and k = 2 . permutations; intersecting families; Erd˝os-Ko-Rado Theorem.
对于满足r≤n的正整数r和n，设P r,n为部分置换族{{(1,x 1)， (2, x 2)，…， (r, x r)}: x 1, x 2，…， x r是{1,2，…的不同元素。， n}}。亚族a1, a2，…， A∩B (cid:54) =∅对于所有A∈A i, B∈A j，其中1≤i (cid:54) = j≤k，则称A k (P r,n)相交。同样，如果a1 a2…， A, k是互不相交的，则称它们为P, r,n的不相交相交亚族。对于不相交的交叉相交亚族a1, a2，…， A k (pn,n)，由AM-GM不等式可得(cid:81) ki =1 |A i |≤(n !/k)本文给出了以下命题的两个证明:(cid:81) ki =1 |A i | = (n !/k) k当且仅当n = 3且k = 2。排列;相交的家庭;Erd˝os-Ko-Rado定理。
{"title":"On Disjoint Cross Intersecting Families of Permutations","authors":"Nuttanon Songsuwan, Supida Sengsamak, Nutchapol Jeerawattana, T. Jiarasuksakun, P. Kaemawichanurat","doi":"10.47443/dml.2022.110","DOIUrl":"https://doi.org/10.47443/dml.2022.110","url":null,"abstract":"For the positive integers r and n satisfying r ≤ n , let P r,n be the family of partial permutations {{ (1 , x 1 ) , (2 , x 2 ) , . . . , ( r, x r ) } : x 1 , x 2 , . . . , x r are different elements of { 1 , 2 , . . . , n }} . The subfamilies A 1 , A 2 , . . . , A k of P r,n are called cross intersecting if A ∩ B (cid:54) = ∅ for all A ∈ A i and B ∈ A j , where 1 ≤ i (cid:54) = j ≤ k . Also, if A 1 , A 2 , . . . , A k are mutually disjoint, then they are called disjoint cross intersecting subfamilies of P r,n . For the disjoint cross intersecting subfamilies A 1 , A 2 , . . . , A k of P n,n , it follows from the AM-GM inequality that (cid:81) ki =1 |A i | ≤ ( n ! /k ) k . In this paper, we present two proofs of the following statement: (cid:81) ki =1 |A i | = ( n ! /k ) k if and only if n = 3 and k = 2 . permutations; intersecting families; Erd˝os-Ko-Rado Theorem.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45509517","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}
Let G ( V, E ) be a simple graph with | V ( G ) | = n and | E ( G ) | = m . If S k ( G ) is the sum of k largest Laplacian eigenvalues of G , then Brouwer’s conjecture states that S k ( G ) ≤ m + k ( k +1)2 for 1 ≤ k ≤ n . The girth of a graph G is the length of a smallest cycle in G . If g is the girth of G , then we show that the mentioned conjecture is true for 1 ≤ k ≤ (cid:98) g − 22 (cid:99) . Wang et al. [ Math. Comput. Model. 56 (2012) 60–68] proved that Brouwer’s conjecture is true for bicyclic and tricyclic graphs whenever 1 ≤ k ≤ n with k (cid:54) = 3 . We settle the conjecture under discussion also for tricyclic graphs having no pendant vertices when k = 3 .
设G (V, E)是一个简单图，其中| V (G) | = n， |e (G) | = m。如果sk (G)是G的k个最大拉普拉斯特征值的和，则Brouwer猜想表明，当1≤k≤n时，sk (G)≤m + k (k +1)2。图G的周长是图G中最小环的长度。如果g是g的周长，那么我们证明了上述猜想对于1≤k≤(cid:98) g−22 (cid:99)成立。Wang et al.[数学]第一版。Model. 56(2012) 60-68]证明了当1≤k≤n且k (cid:54) = 3时，Brouwer猜想对双环和三环图成立。对于k = 3时无垂点的三环图，我们也解决了讨论中的猜想。
{"title":"Computing the Sum of k Largest Laplacian Eigenvalues of Tricyclic Graphs","authors":"Pawan Kumar, S. Merajuddin, S. Pirzada","doi":"10.47443/dml.2022.085","DOIUrl":"https://doi.org/10.47443/dml.2022.085","url":null,"abstract":"Let G ( V, E ) be a simple graph with | V ( G ) | = n and | E ( G ) | = m . If S k ( G ) is the sum of k largest Laplacian eigenvalues of G , then Brouwer’s conjecture states that S k ( G ) ≤ m + k ( k +1)2 for 1 ≤ k ≤ n . The girth of a graph G is the length of a smallest cycle in G . If g is the girth of G , then we show that the mentioned conjecture is true for 1 ≤ k ≤ (cid:98) g − 22 (cid:99) . Wang et al. [ Math. Comput. Model. 56 (2012) 60–68] proved that Brouwer’s conjecture is true for bicyclic and tricyclic graphs whenever 1 ≤ k ≤ n with k (cid:54) = 3 . We settle the conjecture under discussion also for tricyclic graphs having no pendant vertices when k = 3 .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43979861","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}
In this paper, by utilizing the concept of the energy of a vertex, connections between some vertex-degree-based topological indices (including the general Randi´c index, the first Zagreb index, and the forgotten index) and the energy of graphs are established. Several bounds on the energy of the graphs containing no isolated vertices are also given in terms of the first Zagreb index and the forgotten index. Moreover, bounds on the normalized Laplacian energy in terms of two particular cases of the general Randi´c index are obtained.
{"title":"Some Degree-Based Topological Indices and (Normalized Laplacian) Energy of Graphs","authors":"Zimo Yan, Xie Zheng, Jianping Li","doi":"10.47443/dml.2022.059","DOIUrl":"https://doi.org/10.47443/dml.2022.059","url":null,"abstract":"In this paper, by utilizing the concept of the energy of a vertex, connections between some vertex-degree-based topological indices (including the general Randi´c index, the first Zagreb index, and the forgotten index) and the energy of graphs are established. Several bounds on the energy of the graphs containing no isolated vertices are also given in terms of the first Zagreb index and the forgotten index. Moreover, bounds on the normalized Laplacian energy in terms of two particular cases of the general Randi´c index are obtained.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41492917","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}
Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, L'aszlo Lov'asz and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with this property. We strengthen this result by showing that cycles and complete graphs (with all edges considered bidirected) are the only directed graphs with this property.
{"title":"On the Last New Vertex Visited by a Random Walk in a Directed Graph","authors":"Calum Buchanan, P. Horn, Puck Rombach","doi":"10.47443/dml.2022.158","DOIUrl":"https://doi.org/10.47443/dml.2022.158","url":null,"abstract":"Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, L'aszlo Lov'asz and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with this property. We strengthen this result by showing that cycles and complete graphs (with all edges considered bidirected) are the only directed graphs with this property.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46068065","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}
A “harmonic variant” of Zeilberger’s algorithm is utilized to improve upon the results introduced by Wang and Chu [ Ramanujan J. 52 (2020) 641–668]. Wang and Chu’s coefficient-extraction methodologies yielded evaluations for Ramanujan-like series involving summand factors of the form H 3 n +3 H n H (2) n +2 H (3) n , where H n denotes a harmonic number and H ( x ) n is a generalized harmonic number. However, it is unclear as to how Wang and Chu’s techniques could be applied to improve upon such results by separately evaluating the series obtained upon the expansion of the summands according to the terms of the factor H 3 n +3 H n H (2) n +2 H (3) n . In this note, we succeed in applying Zeilberger’s algorithm toward this problem, providing explicit evaluations for the series with a factor of the form H (3) n obtained from the aforementioned expansion. Our approach toward generalizing Zeilberger’s algorithm to non-hypergeometric expressions may be applied much more broadly. The series obtained by replacing H (3) n with H (2) n were highlighted as especially beautiful motivating examples in Wang and Chu’s article. These H (2) n -series motivate our main results, which are natural higher-order extensions of these H (2) n -series.
{"title":"Applications of Zeilberger’s Algorithm to Ramanujan-Inspired Series Involving Harmonic-Type Numbers","authors":"J. Campbell","doi":"10.47443/dml.2022.050","DOIUrl":"https://doi.org/10.47443/dml.2022.050","url":null,"abstract":"A “harmonic variant” of Zeilberger’s algorithm is utilized to improve upon the results introduced by Wang and Chu [ Ramanujan J. 52 (2020) 641–668]. Wang and Chu’s coefficient-extraction methodologies yielded evaluations for Ramanujan-like series involving summand factors of the form H 3 n +3 H n H (2) n +2 H (3) n , where H n denotes a harmonic number and H ( x ) n is a generalized harmonic number. However, it is unclear as to how Wang and Chu’s techniques could be applied to improve upon such results by separately evaluating the series obtained upon the expansion of the summands according to the terms of the factor H 3 n +3 H n H (2) n +2 H (3) n . In this note, we succeed in applying Zeilberger’s algorithm toward this problem, providing explicit evaluations for the series with a factor of the form H (3) n obtained from the aforementioned expansion. Our approach toward generalizing Zeilberger’s algorithm to non-hypergeometric expressions may be applied much more broadly. The series obtained by replacing H (3) n with H (2) n were highlighted as especially beautiful motivating examples in Wang and Chu’s article. These H (2) n -series motivate our main results, which are natural higher-order extensions of these H (2) n -series.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47714167","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}
Ortrud R. Oellermann received an M.Sc. in mathematics from the University of Natal, South Africa in 1983 and a Ph.D. in mathematics from Western Michigan University, USA in 1986. She taught at several universities, but the majority of her academic career was spent at the University of Winnipeg, Canada, where she served from July 1996 until August 31, 2021, when she retired as a professor. She is currently an adjunct professor of mathematics at both the University of Winnipeg and the University of Victoria, Canada. Professor Oellermann was honoured with a Professor Emerita title from the University of Winnipeg in June 2022. Throughout her career she held research grant funding from research funding agencies such as the Office of Naval Research (USA), the National Research Foundation (South Africa) and NSERC (Canada). To date she has 85 co-authors of which 22 are former research students or post-doctoral fellows. She is currently one of four editors-in-chief of the Bulletin of the Institute of Combinatorics and its Applications. Previously she served on the editorial boards of Ars Combinatoria and Utilitas Mathematics. Professor Oellermann has received several medals, including the Hall Medal from the Institute of Combinatorics and its Applications in 1995. She was an elected member of the board of directors of the Canadian Mathematical Society (July 2001 June 2005) and the executive committee of the Discrete Mathematics activity group of the Society for Industrial and Applied Mathematics (January 2006 December 2007). Professor Oellermann also served as an academic consultant for the Cambridge University Press monograph “Topics in Structural Graph Theory” edited by Lowell W. Beineke and Robin J. Wilson.
Ortrud R. Oellermann于1983年在南非纳塔尔大学获得数学硕士学位，1986年在美国西密歇根大学获得数学博士学位。她曾在多所大学任教，但她的大部分学术生涯是在加拿大温尼伯大学度过的，从1996年7月到2021年8月31日，她在那里担任教授，退休。她目前是加拿大温尼伯大学和维多利亚大学的兼职数学教授。Oellermann教授于2022年6月被温尼伯大学授予名誉教授称号。在她的职业生涯中，她获得了研究资助机构的研究资助，如海军研究办公室(美国)，国家研究基金会(南非)和NSERC(加拿大)。到目前为止，她有85位合著者，其中22位是以前的研究生或博士后。她目前是《组合学及其应用研究所公报》的四位主编之一。此前，她曾在Ars Combinatoria和Utilitas Mathematics的编辑委员会任职。Oellermann教授曾获得多项奖章，包括1995年由组合学及其应用研究所颁发的霍尔奖章。她是加拿大数学学会董事会的当选成员(2001年7月2005年6月)和工业与应用数学学会离散数学活动小组的执行委员会(2006年1月2007年12月)。Oellermann教授还担任剑桥大学出版社专著“结构图论主题”的学术顾问，该专著由Lowell W. Beineke和Robin J. Wilson编辑。
{"title":"An Interview With Ortrud Oellermann","authors":"Akbar Ali","doi":"10.47443/dml.2022.i1","DOIUrl":"https://doi.org/10.47443/dml.2022.i1","url":null,"abstract":"Ortrud R. Oellermann received an M.Sc. in mathematics from the University of Natal, South Africa in 1983 and a Ph.D. in mathematics from Western Michigan University, USA in 1986. She taught at several universities, but the majority of her academic career was spent at the University of Winnipeg, Canada, where she served from July 1996 until August 31, 2021, when she retired as a professor. She is currently an adjunct professor of mathematics at both the University of Winnipeg and the University of Victoria, Canada. Professor Oellermann was honoured with a Professor Emerita title from the University of Winnipeg in June 2022. Throughout her career she held research grant funding from research funding agencies such as the Office of Naval Research (USA), the National Research Foundation (South Africa) and NSERC (Canada). To date she has 85 co-authors of which 22 are former research students or post-doctoral fellows. She is currently one of four editors-in-chief of the Bulletin of the Institute of Combinatorics and its Applications. Previously she served on the editorial boards of Ars Combinatoria and Utilitas Mathematics. Professor Oellermann has received several medals, including the Hall Medal from the Institute of Combinatorics and its Applications in 1995. She was an elected member of the board of directors of the Canadian Mathematical Society (July 2001 June 2005) and the executive committee of the Discrete Mathematics activity group of the Society for Industrial and Applied Mathematics (January 2006 December 2007). Professor Oellermann also served as an academic consultant for the Cambridge University Press monograph “Topics in Structural Graph Theory” edited by Lowell W. Beineke and Robin J. Wilson.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42144537","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}
The Boole’s additive combinatorics formula is given by n (cid:88) k =0 ( − 1) n − k (cid:32) n k (cid:33) k m = 0 if m < n, n ! if m = n. A new proof of this formula is presented in this paper.
《布尔additive combinatorics是赐予由n (cid的公式:88)k = 0 (n−1)k(−cid 33: 32) n k (cid) k m =0如果m < n, n !如果m = n. A新的公式证明在这张纸上。
{"title":"A New Proof of Boole’S Additive Combinatorics Formula","authors":"Necdet Batır, S. Atpinar","doi":"10.47443/dml.2022.109","DOIUrl":"https://doi.org/10.47443/dml.2022.109","url":null,"abstract":"The Boole’s additive combinatorics formula is given by n (cid:88) k =0 ( − 1) n − k (cid:32) n k (cid:33) k m = 0 if m < n, n ! if m = n. A new proof of this formula is presented in this paper.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46250372","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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