A proof of the \(q,t\)-square conjecture
type: Item
instance of: scholarly article
title: A proof of the \(q,t\)-square conjecture
zbMATH Open document ID: 1110.05098
published in: Journal of Combinatorial Theory. Series A
publication date: 2006-12-07T00:00:00Z
review text: The authors prove a combinatorial formula conjectured by \textit{N. A. Loehr} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 359, No. 2, 649--669 (2007; Zbl 1107.05098)] for the coefficient of the sign charater in \(\nabla(p_n)\), where \(\nabla\) is the Bergeron-Garsia nabla operator [\textit{F. Bergeron} and \textit{A. M. Garsia}, Algebraic methods and \(q\)-special functions. CRM Proc. Lect. Notes 22, 1--52 (1999; Zbl 0947.20009)] and \(p_n\) the power sum symmetric function. The formula is analogous to the \((q,t)\)-Catalan theorem, which givs a combinatorial formula for \(\nabla(e_n)\), where \(e_n\) is the elementary symmetric function. The combinatorial formula in the \((q,t)\)-Catalan theorem enumerates Dyck paths (lattice paths in the square which stay on or above the main diagonal) weighted by certain statistics. The analogous formula for the \((q,t)\)-square conjecture enumerates all lattice paths in the square weighted by similar statistics.
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 33D52
zbMATH DE Number: 5078223
zbMATH Keywords: square lattice paths
zbMATH Keywords: \((q,t)\)-Catalan numbers
zbMATH Keywords: Bergeron-Garsia nabla operator
zbMATH Keywords: plethysm
zbMATH Keywords: Macdonald polynomials
zbMATH Keywords: symmetric function
author: Mahir Bilen Can
author: Nicholas A. Loehr
reviewed by: David Grabiner
MaRDI profile type: Publication
full work available at URL: j.jcta.2006.01.002
OpenAlex ID: W2085858850
Wikidata QID: Q123154639
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DOI: 10.1016/J.JCTA.2006.01.002
description: scientific article; zbMATH DE number 5078223