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# Journal of Combinatorics

## Volume 2 (2011)

### Number 2

### The case $k=2$ of the Shuffle Conjecture

Pages: 193 – 229

DOI: http://dx.doi.org/10.4310/JOC.2011.v2.n2.a2

#### Authors

#### Abstract

It was conjectured in [5] and proved by Mark Haiman in [13] that theFrobenius Characteristic of the $S_n$ Module of Diagonal Harmonics isnone other than $\nabla e_n$. Here $\nabla$ is the symmetric functionoperator introduced in [1] with eigen-functions the modified Macdonaldbasis $\{\TH_\mu\}_\mu$. The Shuffle Conjecture [12] expresses thescalar product $\LL \nabla e_n\scs h_{\mu_1} h_{\mu_2}\cdotsh_{\mu_k}\RR$ as a weighted sum of Parking Functions on the $n\times n$lattice square which are shuffles of $k$ increasing words. In [10]Jim Haglund succeeded in proving the $k=2$ case of this conjecture. Inthis paper we give a new and more direct proof of the combinatorialpart of Haglund’s argument and obtain a substantial reduction in thecomplexity of the symmetric function part.