A three shuffle case of the compositional parking function conjecture

TL;DR

This paper proves a specific case of the longstanding shuffle parking function conjecture, connecting algebraic polynomials with combinatorial parking functions and their statistics, advancing the understanding of Macdonald polynomials and related conjectures.

Contribution

It establishes a new case of the shuffle parking function conjecture involving Dyck paths, Hall-Littlewood polynomials, and Macdonald operators, expanding previous results.

Findings

Proves a specific shuffle parking function case involving Dyck paths and compositions.

Connects algebraic polynomials with combinatorial parking functions and statistics.

Includes previous results as special cases, supporting the conjecture's broader validity.

Abstract

We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> 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 of size a + b + c and have a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp(1) is a rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator introduced in [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of [14]. It includes as special cases all previous results connected with this conjecture such as the q, t-Catalan [3] and the Schroder and h, h results of Haglund in [12] as well as their compositional refinements recently obtained in [9] and [10]. It also confirms the possibility that the approach adopted in [9] and [10] 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 in [15].