2024/08/13 by Spencer Backman, Backman, Spencer, Cole Charbonneau +9
Computer Science · Mathematics · #05A15 #05A19 #05C57 #Advanced Graph Theory Research #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2408.06923
openalex publication_date 2024/08/13 · openalex created_date 2024/09/11 · openalex updated_date 2026/07/28
For 0≤ k≤ n-1, we introduce a family of k-skeletal paths which are counted by the n-th Catalan number for each k, and specialize to Dyck paths when k=n-1. We similarly introduce k-skeletal parking functions which are equinumerous with the spanning trees on n+1 vertices for each k, and specialize to classical parking functions for k=n-1. The preceding constructions are generalized to paths lying in a trapezoid with base c > 0 and southeastern diagonal of slope 1/m; c and m need not be integers. We give bijections among these families when k varies with m and c fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.