A three shuffle case of the compositional parking function conjecture

Abstract

We prove here that the polynomial 〈∇Cp1,eahbhc〉q,t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p⊨a+b+c and has a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp1 is a rescaled Hall–Littlewood polynomial and ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of Haglund et al. This result includes as special cases all previous results connected with the shuffle conjecture such as the q,t-Catalan, Schröder and two shuffle results of Haglund as well as their compositional refinements recently obtained by the authors. It also confirms the possibility that the approach adopted by the authors has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki.