Counting Restricted Set Partitions with Hypergeometrics

Explore the intricacies of counting restricted set partitions using hypergeometric functions in this 46-minute lecture by Karol Penson from the Institut des Hautes Etudes Scientifiques (IHES). Delve into the exponential generating function of Bell numbers and investigate properties of integers describing restricted partitions. Examine two deformation methods of the exponent in the generating function: subtracting finite terms and retaining only finite terms. Discover how these deformations relate to restricted partitions without certain box sizes or with only specific box sizes. Learn about the application of multi-variable extensions of Hermite polynomials to obtain closed-form expressions for restricted Bell numbers using generalized hypergeometric functions. Gain insights into the efficient handling of these formulas using Computer Algebra Systems, while also considering the limitations of this approach.