Journal of Combinatorics

Volume 10 (2019)

Number 4

Special Issue in Memory of Jeff Remmel, Part 2 of 2

Guest Editor: Nicholas A. Loehr

Some new symmetric function tools and their applications

Pages: 655 – 674

DOI: https://dx.doi.org/10.4310/JOC.2019.v10.n4.a3

Authors

A. Garsia (University of California at San Diego)

J. Haglund (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

M. Romero (University of California at San Diego)

Abstract

We prove a technical identity involving the $\Delta$ operator from MacDonald polynomial theory, which allows us to transform expressions involving the $\Delta$ operator and the Hall scalar product into other such expressions. We show how our technical identity, although following easily from the well-known Koornwinder–MacDonald reciprocity theorem, contains as special cases several identities occuring in the literature, proved there by more complicated arguments. We also show how our identity can be used to obtain some new expressions for the $q, t$-Narayana numbers introduced by Dukes and Le Borgne, as well as new identities involving the $\Delta$ operator and the power sum symmetric function $p_n$.

Full Text (PDF format)

A. Garsia was supported by NSF grant DMS-1700233.

J. Haglund was supported by NSF grant DMS-1600670.

M. Romero was supported by NSF grant DMS-1362160.

Received 11 July 2018

Published 17 July 2019