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

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.

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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