A refinement of the Shuffle Conjecture with cars of two sizes and $t=1/q$

Abstract

The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as ∇e n , h µ where ∇ is the Macdonald polynomial eigen-operator of Bergeron and Garsia and h µ is the homogeneous basis indexed by µ = (µ 1 , µ 2 , . . . , µ k ) n. The combinatorial side q,tenumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 123 · · · n of respective lengths µ 1 , µ 2 , . . . , µ k . It can be shown that for t = 1/q the symmetric function side reduces to a product of q-binomial coefficients and powers of q. This reduction suggests a surprising combinatorial refinement of the general Shuffle Conjecture. Here we prove this refinement for k = 2 and t = 1/q. The resulting formula gives a q-analogue of the well studied Narayana numbers.

Combinatorics of the Delta conjecture at q=-1

“…In this section we discuss the schedule formula for the combinatorics of the valley Delta conjecture proved by Haglund and Sergel in [13]. Their formula is an extension of the first work on schedule numbers by Hicks in her thesis [16].…”

“…n , then σ0 is never decorated. Definition 3.7 (Hicks [16]). For τ ∈ S • n define its schedule numbers sched(τ ) = (si) 1≤i≤n as follows.…”