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- LOUIS W. SHAPIRO
- 2010

In this note a combinatorial correspondence is used to prove that the number of positive definite, tridiagonal, integral matrices of determinant 1 whose sub and super diagonals consist solely of ones is C " = (2 " ")/(n + 1). The correspondence is then further used to count such matrices by trace and also by number of ones on the main diagonal. Other… (More)

Shapiro, L.

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the… (More)

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… (More)