Total Positivity Of Combinatorial Triangles

Total positivity is an important and powerful concept that arises often in various branches of mathematics, statistics, probability, mechanics, economics, and computer science. It is also realized that TP matrices are quite ubiquitous in combinatorics. The object of this thesis is to study various positivity properties of recursive matrices and Riordan arrays in a unified approach. The thesis is organized as follows.In the first part we present some sufficient conditions for total positivity of recursive matrices, which are closely related to tridiagonal matrices. As applications, we show that the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the Catalan triangles and the central binomial triangle are TP. As consequences, the corresponding Catalan-like numbers, including the Catalan numbers, the Bell numbers, the large Schroder numbers and the central binomial coefficients, form a log-convex sequence. We also point out that our results can be carried over verbatim to their q-analogues. Other types of recursive matrices are discussed at the end of this part.The second part is devoted to various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column and the log-concavity or even PF property of each row. Applications to two classes of special interesting Riordan arrays are considered. One is Aigner-Riordan arrays, the other is consistent Riordan arrays. As consequences, we prove the total positivity of the Pascal triangle and the Catalan triangle of Shapiro, as well as that of the ballot table, the large and the little Schroder triangles. In addition, the Catalan numbers, the large and little Schroder numbers are proved to be log-convex.Finally, in the third part, we present combinatorial proofs for the total positivity of recursive matrices. Additionally we give a combinatorial interpretation for Riordan-like matrices, which contain both recursive matrices and Riordan arrays, in terms of weighted Lukasiewicz paths. Furthermore, a combinatorial proof of the log-convexity of Aigner-Catalan-Riordan numbers is provided.