Boundaries of the Generalised Pascal Triangles

Abstract

Many classical triangular arrays of combinatorial numbers (Pascal, q-Pascal, Euler, Stirling) are representable in terms of path-counting on the two-dimensional lattice in the quadrant with weighted edges. Enumeration of paths may be associated with some combinatorial structures (e.g. permutations, partitions) of variable size. With each such triangle one associates a class of random walks, which can also be understood as series of dependent Bernoulli trials with a natural sufficiency property. Points of the boundary correspond to ergodic random walks. The talk will focus on the general properties of the boundary and its structure for certain triangles that support random walks with transition probabilities of generalised factorial form.