Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

Abstract

We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials indexed by hooks. Our combinatorial formula relates Macdonald cumulants to the generating function of $G$-parking functions, or equivalently to a certain specialization of the Tutte polynomials.

25 pages, 6 figures

References in corpus (6)

• Fluctuations of particle systems determined by Schur generating functions
• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• Toric braids and $(m,n)$-parking functions
• Compositional (km,kn)-Shuffle Conjectures
• Gaussian fluctuations of Jack-deformed random Young diagrams
• Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture

Cited by in corpus (2)

• A positive combinatorial formula for symplectic Kostka-Foulkes polynomials I: Rows
• LLT cumulants of unicellular Young diagrams, parking functions and Schur positivity