Some Problems In Algebraic Combinatorics

Algebraic Combinatorics is an important branch of combinatorics. The aim of this thesis is to investigate some classical and important problems in algebraic combinatorics, including Sperner-type problems, the supersolvablity of hyperplane arrangements and combinatorics of Riordan matrices. The main frame of the thesis is as follows.In the first part we investigate the generalized Sperner theorem in convex families. Sperner theorem is one of the central results in extremal set theory, and it has been generalized and ex-tended a lot. Akiyama and Frankl conjectured that Sperner theorem holds in convex families. We provide further evidence for the conjecture by exhibiting a number of classical convex families.The second part is devoted to the supersolvability of the hyperplane arrangements. An ar-rangement A is supersolvable if the intersection lattice L(c(A)) of the cone c(A) contains a max-imal chain of modular elements. Stanley showed that a graphical arrangement is supersolvable if and only if the graph is a chordal graph. We consider a generalization of graphical arrangements which are called ?-graphical arrangements. We give a characterization of the supersolvability and freeness (in the sense of Terao) of a ?-graphical arrangement. It is well known that every supersolvable arrangement is free, and every free graphical arrangement is supersolvable. We also provide some conditions on free ?-graphical arrangements.In the third part we discuss the combinatorial properties of row polynomial matrices and the first column of Riordan arrays, including the characterizations, the q-total positivity of the row polynomial matrix, the q-log-convexity of the first column of row polynomial matrix and the q-log-concavity of each row of row polynomial matrix. The row polynomial matrices of Bell-type and Aigner-type Riordan arrays are the emphases of this part. A unified approach to deal with certain row sum problems is provided by means of row polynomial matrices in this part. Barry proposed three conjectures about Hankel determinant evaluations of some series reversions. We settle Barry's three conjectures in a unified approach.