Séminaire Lotharingien de Combinatoire, 80B.41 (2018), 12 pp.
Maciej Dołęga
A Combinatorial Formula for Macdonald Cumulants
Abstract.
Macdonald cumulants are symmetric functions
that generalize Macdonald polynomials. We prove a combinatorial formula for them which
extends the celebrated formula of Haglund for Macdonald
polynomials. We also provide several applications of our formula -- it gives a new, constructive
proof of a strong factorization property
of Macdonald polynomials and 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
links Macdonald cumulants to G-parking
functions of Postnikov and Shapiro.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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