2005/09/12 by William Y. C. Chen, Chen, William Y. C., Nelson Y. Li +6
Computer Science · Mathematics · #05A15 #05A19 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications #Topological and Geometric Data Analysis #math.CO #msc:05A15 #msc:05A19
paper · pdf · doi:10.48550/arxiv.math/0509255
15 pages, 3figures
arxiv created 2005/09/12 · openalex publication_date 2005/09/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1, 4, 42, 43, ...) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1, k, k2, k3, ...) for any k ≥ 2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1, t2+t, (t2+t)2, ...).