Geodesic
A Combinatorial Formula for Macdonald Cumulants
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a40,
     author = {Maciej Do{\l}\k{e}ga},
     title = {A {Combinatorial} {Formula} for {Macdonald} {Cumulants}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a40/}
}
TY  - JOUR
AU  - Maciej Dołęga
TI  - A Combinatorial Formula for Macdonald Cumulants
JO  - Séminaire lotharingien de combinatoire
PY  - 2018
VL  - 80B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a40/
ID  - SLC_2018_80B_a40
ER  - 
%0 Journal Article
%A Maciej Dołęga
%T A Combinatorial Formula for Macdonald Cumulants
%J Séminaire lotharingien de combinatoire
%D 2018
%V 80B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018_80B_a40/
%F SLC_2018_80B_a40
Maciej Dołęga. A Combinatorial Formula for Macdonald Cumulants. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a40/