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

摘要

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 1/n+1 (2n/n). We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of sr,1n-r in the expression Δh_? ?C_α is equal to a weighted sum over decorated Dyck paths.