Monday December 5th, 2022 | |
| Time: | 2:30 PM |
| Title: | Eigenvalues for the random-to-below shuffle |
| Speaker: | Nadia Lafrenière, Dartmouth College |
| Abstract: | |
| The random-to-below shuffle of a deck of cards consists of removing any card at random (with uniform probability), and inserting it anywhere below (with uniform probability). A surprising fact is that the eigenvalues of its transition matrix are all rational and positive. This is unexpected for a non-symmetric matrix, and suggests some combinatorial interpretation. I’ll discuss an expression of the eigenvalues in terms of lacunar sets, namely sets with no consecutive integers, and make connections with the well studied top-to-random shuffle. I'll then suggest how this may be extended to standard tableaux. I’ll finally describe a more straightforward way to compute a stopping time for this shuffle, answering the question of how long we would need to shuffle to get a well-mixed deck using this shuffling technique. This is joint work with Darij Grinberg. | |