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

TL;DR

This paper proves a key case of the Delta conjecture by establishing a combinatorial interpretation for a four-variable Catalan polynomial related to decorated Dyck paths.

Contribution

It provides the first proof of the compositional version of the four-variable Delta conjecture, linking algebraic coefficients to combinatorial path statistics.

Findings

Proves the compositional Delta conjecture for four-variable Catalan polynomial.

Establishes a combinatorial formula involving decorated Dyck paths.

Connects algebraic coefficients to weighted sums over combinatorial objects.

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 $\frac{1}{n+1}\binom{2n}{n}$. 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.