A proof of the Square Paths Conjecture

The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related ∇ 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 ∇e_n to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for ∇p_n which is known as the Square Paths Conjecture. Haglund and Loehr (2005) introduced the notion of schedules to enumerate parking functions with a fixed set of cars in each diagonal. In this paper, we extend the notion of schedules and some related results of Hicks (2013) to labeled square paths. We then apply our new results to prove the Square Paths Conjecture.