MIT Combinatorics Seminar

Chess tableaux and chess problems

Timothy Chow

Wednesday, October 20, 2004    4:15 pm    Room 2-338


A chess tableau is a standard Young tableau (SYT) in which orthogonally adjacent entries have opposite parity. Remarkably, the number of 3xn chess tableaux is the same as several other quantities: the number of 3x(n-1) nonconsecutive tableaux (SYT in which i and i+1 never appear in the same row), the Charney-Davis statistic of a 3xn shape, and the number of Baxter permutations of n. Yet there is no obvious bijection between any two of these. Our main result is a pleasant but mysterious bijection between chess tableaux and nonconsecutive tableaux with three rows. Bijections with the Charney-Davis statistic remain an open problem.

Our original motivation was, appropriately enough, composing chess problems. In the last part of the talk we present and explain two chess problems (one by Noam Elkies) that are related to chess tableaux. The problems have the same flavor as the chess problems in Stanley's book EC2.

The definition of a Young tableau and (for the last part of the talk) knowledge of how chess pieces move are sufficient background for the talk; other terminology will be explained.

This is joint work with Ken Fan and Henrik Eriksson.