Matrix identities on weighted partial Motzkin paths

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 path...

Full description

Saved in:
Bibliographic Details
Published in:European journal of combinatorics 2007-05, Vol.28 (4), p.1196-1207
Main Authors: Chen, William Y.C., Li, Nelson Y., Shapiro, Louis W., Yan, Sherry H.F.
Format: Article
Language:English
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 , … ) .
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2006.02.005