New combinatorial formulations of the shuffle conjecture

Authors

Nicholas A. Loehr
Elizabeth Niese, Marshall UniversityFollow

Document Type

Article

Publication Date

4-2014

Abstract

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DH_n, which is the symmetric function∇(e_n). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇^m(e_n). We also give new versions of some recently proposed combinatorial formulas for ∇(C_α) and ∇(s_{(k,1^(n−k))}), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund's bounce statistic.

Comments

This is the authors’ submitted manuscript.

The version of record is available from the publisher at http://dx.doi.org/10.1016/j.aam.2013.12.003.

Copyright © 2014 Elsevier Inc. All rights reserved.

Recommended Citation

Loehr, N. A., & Niese, E. (2014). New combinatorial formulations of the shuffle conjecture. Advances in Applied Mathematics, 55, 22-47.