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Matrix Identities on Weighted Partial Motzkin Paths | | William Y.C. Chen
; Nelson Y. Li
; Louis W. Shapiro
; Sherry H.F. Yan
; | | Date: |
12 Sep 2005 | | Subject: | Combinatorics MSC-class: 05A15, 05A19 | math.CO | | 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 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 $(1, t^2+t, (t^2+t)^2, ...)$. | | Source: | arXiv, math.CO/0509255 | | Services: | Forum | Review | PDF | Favorites | |
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