Total Positivity And Asymptotic Normality Of Combinatorial Matrices

Combinatorial matrices play a fundamental role in enumerative combinatorics.In the matrix theory,total positivity and asymptotic normality are two important concepts.The objective of this thesis is to study the total positivity and asymptotic normality of combinatorial matrices.The thesis is organized as follows.In the first part we study the total positivity of carry matrices.The carry matrix is the transition probability matrix of the Markov chain of carry process.Diaconis and Fulman conjectured that the carry matrices are totally positive.We show that the carry matrices are the submatrices of a generalized Hurwitz matrix.By proving the total positivity of the infinite generalized Hurwitz matrix,we obtain that the carry matrix is totally positive.In the second part we study the asymptotic normality of the descent matrices of stack sortable permutations.Stack sortable permutations are important research objects in permutation patterns.We study a class of matrices whose entries are the number of t-stack sortable n-permutations with k-1 descents.When t=1 and n-1,the matrices correspond to the Narayana triangle and the Eulerian triangle respectively.We show that for the cases t=1,2,n-2 and n-1,the descent matrices of t-stack sortable n-permutations are asymptotically normal,which in particular gives an affirmative answer to Shapiro’s question on the asymptotic normality of Narayana numbers.In the third part we study the total positivity of ECO matrices.ECO matrices play an important unifying role in combinatorial enumeration.Many combinatorial matrices are ECO matrices such as Riordan arrays,exponential Riordan arrays and recursive matrices.We establish a criterion for the total positivity of ECO matrices.It provides a unifying approach to the total positivity of these combinatorial matrices.In particular,we present a criterion for the total positivity of the Riordan arrays.