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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