A proof of the Square Paths Conjecture

摘要

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related del operator of Bergeron and Garsia (1999) to basic symmetric functions. The first discovery of this type was the (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005), which relates the expression del(en) to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for del(Pn) which is known as the Square Paths Conjecture.