Shuffling Cards, Adding Numbers, and Symmetric Functions
Problems of riffle shuffling cards don't seem to have much to do with problems of analyzing "carries" when adding two integers base 10. Both subjects seem remote from modern algebraic combinatorics, where things like quasi-symmetric functions are all the rage. In this talk, I will review these three subjects and show that they are intimately related.
For shuffling, I will review the "seven shuffles" theorem (with David Bayer) and extensions to modern casino shelf-shufflers (work with Susan Holmes and Jason Fulman). For carries, I will review the work of Knuth and Holte on the process of carries when m integers are added base 6. For symmetric functions, I will explain how much of combinatorics has been unified---theories of permutation enumeration, partions, and partially ordered sets are included.
Finally, in work with Jason Fulman, I will reveal that "it's all just shuffling."
Persi Diaconis, Stanford University