Daniel Herden, Baylor University

Abstract

Starting point of our talk will be a short discussion of the classical 15 puzzle. It turns out that similar problems about sliding numbered tiles around in a box are of enormous relevance to combinatorial algebra in the context of Young tableaux. In particular, we are going to present "The Tableau Reconstruction Problem," posed by Monks (2009). Starting with a standard Young tableau T of size n, i.e., an arrangement of n numbered tiles, a 1-minor of T is a tableau obtained by first deleting any tile of T, and then sliding the remaining tiles into position and renumbering them according to some rules. This can be iterated to arrive at the set of k-minors of T. The problem is this: given k, what are the values of n such that every tableau of size n can be reconstructed from its set of k-minors? For k=1, the problem was recently solved by Cain and Lehtonen. In this talk, we discuss the problem for k=2, proving the sharp lower bound n ≥ 8. In the case of multisets of k-minors, we also give a lower bound for arbitrary k, as a first step toward a sharp bound in the general multiset case.

Link to video: https://vimeo.com/1061088990?share=copy

Presented Via Zoom: https://rutgers.zoom.us/j/91865817691

Password: 6564120420

For further information see: https://sites.math.rutgers.edu/~zeilberg/expmath/