Iterative Families of Partitions in Non-commutative Probability Theory

Learn about semi-multiplicative functions over non-crossing partitions and their applications in a 26-minute lecture from the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability" at the Erwin Schrödinger International Institute. Explore how these functions naturally act on sequences of multilinear functionals in non-commutative probability spaces, enabling systematic study of combinatorial transitions between moments and cumulants. Discover how the group of semi-multiplicative functions over NC functions as a subgroup within a larger structure involving the incidence algebra of all partitions P, incorporating classical moment-cumulant formulas. Examine various iterative families S that possess sufficient structure to support semi-multiplicative functions, expanding understanding of these mathematical relationships.