Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means ...Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.展开更多
Let p be an odd prime and let a,m ∈ Z with a > 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by ...Let p be an odd prime and let a,m ∈ Z with a > 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0<k<pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.展开更多
A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, comput...A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, computational search and symbolic dynamics, the mathematical structure underlying the time series generated from the elementary cellular automaton of rule 56 is analyzed and its complexity is determined, in which the Dyck language and Catalan numbers emerge naturally.展开更多
A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and...A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.展开更多
文摘Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.
基金supported by National Natural Science Foundation of China (Grant No.10871087)the Overseas Cooperation Fund of China (Grant No.10928101)
文摘Let p be an odd prime and let a,m ∈ Z with a > 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0<k<pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.
基金This work is supported by the Special Funds for Major State Basic Research Project.
文摘A new approach to study the evolution complexity of cellular automata is proposed and explained thoroughly by an example of elementary cellular automaton of rule 56. Using the tools of distinct excluded blocks, computational search and symbolic dynamics, the mathematical structure underlying the time series generated from the elementary cellular automaton of rule 56 is analyzed and its complexity is determined, in which the Dyck language and Catalan numbers emerge naturally.
文摘A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.
