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# Journal of Combinatorics

## Volume 10 (2019)

### Number 4

### Special Issue in Memory of Jeff Remmel, Part 2 of 2

Guest Editor: Nicholas A. Loehr

### A proof of the 4-variable Catalan polynomial of the Delta conjecture

Pages: 599 – 632

DOI: https://dx.doi.org/10.4310/JOC.2019.v10.n4.a1

#### Author

#### Abstract

In *The Delta Conjecture* [HRW15], Haglund, Remmel and Wilson introduced a four variable $q, t, z, w$-Catalan polynomial, so named because the specialization of this polynomial at the values $(q, t, z, w) = (1, 1, 0, 0)$ is equal to the Catalan number$\frac{1}{n+1} \left (\begin{smallmatrix} 2n \\ n \end{smallmatrix}\right ) $.We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $\nabla_{h_{\ell}} \nabla C_{\alpha}$ is equal to a weighted sum over decorated Dyck paths.

This research is supported by NSERC.

This paper is dedicated to the memory of Jeffery Remmel (1948–2017).

Received 22 February 2018

Published 17 July 2019