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

Abstract

In The Delta Conjecture (arxiv:1509.07058), 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 $\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$. 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 ${\Delta }_{h_{\ell }}\nabla C_{\alpha }$ is equal to a weighted sum over decorated Dyck paths.