UC San Diego

The symmetric function operator, nabla, introduced by Bergeron and Garsia

(1999), has many astounding combinatorial properties. The (recently proven)

Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005) relates

nabla en to parking functions. The rational Compositional Shuffle Conjecture

of the author, Bergeron, Garsia, and Xin (2015) relates a whole family of operators

(closely linked to nabla) to rational parking functions. Loehr and

Warrington (2007) conjectured a relationship between nabla pn and preference functions. We

prove this conjecture and provide another combinatorial interpretation in terms

of parking functions. This new formula reveals a connection between nabla pn and an

operator appearing in the rational Compositional Shuffle Conjecture at t = 1/q.