Combinatorial interpretations of Hankel matrices and further combinatorial uses of Riordan group methods.

摘要

Random walk enumeration is a key concept in enumerative combinatorics. Several well known sequences have been used to enumerate various random walks such as the Catalan numbers, Motzkin numbers and Schröder numbers. These enumerations have applications in queuing theory, sorting and searching algorithms. Hankel matrices have applications in complex analysis and approximation theory. This research will address the Hankel matrices of the well known combinatorial sequences related to the Schröder numbers. Motivated by the Schröder numbers results, the research solves other related enumeration problems. For the sequences that arise, the research provides additional combinatorial settings which yield these numbers. In addition to the enumeration problems, the research explores the relationship between Hankel matrices and disjoint path systems.