**Matrix Identities on**

Weighted Partial Motzkin Paths

William Y.C. Chen$^1$, Nelson Y. Li$^2$, Louis W. Shapiro$^3$ and Sherry H. F. Yan$^4$\
\[2mm\] $^{1,2,4}$Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China\
\[2mm\] $^3$Department of Mathematics, Howard University, Washington, DC 20059, USA\
\[2mm\] $^1$chen@nankai.edu.cn, $^2$nelsonli@eyou.com, $^3$lshapiro@howard.edu, $^4$huifangyan@eyou.com\

[**Abstract.**]{} We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence $(1, 4,
4^2, 4^3, \ldots)$ 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 $(1, k, k^2, k^3, \ldots)$ for any $k \geq 2$. 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 $(1, t^2+t, (t^2+t)^2, \ldots)$.

[Key words]{}: Catalan number, Schröder number, Dyck path, Motzkin path, partial Motzkin path, free Motzkin path, weighted Motzkin path, Riordan array

[AMS Mathematical Subject Classifications]{}: 05A15, 05A19.

[Corresponding 