A Parking Function Bijection supporting the Haglund-Morse-Zabrocki\n Conjectures

2012/10/09 by Angela M. Hicks, Hicks, Angela
Mathematics · #05A19 (Secondary) #05E05 (Primary) 05E10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1210.2705

openalex publication_date 2012/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The shuffle conjecture expresses a relationship between parking functions,\ndiagonal harmonics, and the Bergeron-Garsia \∇ operator. Recent\nconjectures about a family of modified Hall-Littlewood operators made by\nHaglund, Morse, and Zabrocki sharpen the shuffle conjecture and suggest a\nvariety of combinatorial properties of parking functions. In particular, their\nconjectures combined with previously established commutativity laws of the\nHall-Littlewood operators, suggest the existence of certain bijections relating\nparking functions with different diagonal compositions. In this paper we\nformulate a conjecture which yields an algorithm for the construction of these\nbijections, prove a special case, and give some applications.\n

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