Unimodality Properties Of Sequences In Combinatorial Triangles And Row Polynomial Triangles

Distribution property is an important topic in combinatorics,appearing in combinatoric-s,analysis,algebra,number theory,probability statistics,computer science,economics and other subjects.Unimodality property,which includes unimodality,convexity(concavity),log-convexity(log-concavity),k-q-convexity(k-q-concavity),is one of the basic distribution properties.And with the development of combinatorics,unimodality property has attracted more and more attention.Combinatorial triangle is one of the primary topics in combinatorics.Combinatorial triangles in combinatorics include Pascal triangle,Catalan triangle,etc.There are some results on the unimodality property of the sequences in these combinatorial triangles and their corresponding row polynomial triangles.For example,the sequence of each row in Pascal triangle is log-concave,and the sequence of each row in Pascal row polynomial triangle is q-log-concave.Therefore,it is important to study the unimodality property of sequences in combinatorial triangle and its corresponding row polynomial triangle.In this paper,the unimodality property of sequences in combinatorial triangle and its corresponding row polynomial triangle is mainly studied.The main frame of the paper is as follows.The first part mainly discusses the q-convexity(q-concavity),the 2-q-convexity(2-q-concavity)and the q-increasing(q-decreasing)of the sequences(R_n0+id,k0+i?(q)_i?0 in the row polynomial triangle.Firstly,the expression of a sequence locating on a transversal of a row polynomial triangle corresponding to a combinatorial triangle is given by using the parameters n₀,k₀,d,?.The sequences are divided into finite sequences and infinite sequences.And then we obtain the unimodality property of different sequences locating on transversals.Finally,as an application,we show the q-convexity(q-concavity),the q-increasing(q-decreasing)and the 2-q-convexity(2-q-concavity)of a sequence locating on a transversal of the row polynomial triangles corresponding to the Pascal triangle and three Catalan triangles.The second part mainly discusses the 2-convexity(2-concavity)of sequences in three Catalan triangles.The expression of sequence locating on a transversal in these combinatorial triangles is also given by using the parameters.We get the unimodality property of these sequences locating on all kinds of transversals by algebraic methods.