Matrix identities on weighted partial Motzkin paths

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 , 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 , … ) .