Matrix identities on weighted partial Motzkin paths

Authors:
William Y. C. Chen;Nelson Y. Li;Louis W. Shapiro;Sherry H. F. Yan
Affiliations:
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China;Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China;Department of Mathematics, Howard University, Washington, DC 20059, USA;Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China
Venue:
European Journal of Combinatorics
Year:
2007

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Abstract

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1,4,4^2,4^3,...) 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,k^2,k^3,...) for 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,t^2+t,(t^2+t)^2,...).