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A Dyck path of length 2n is a path in two-space from (0, 0) to (2n, 0) which uses only steps (1, 1) (northeast) and (1, −1) (southeast). Further, a Dyck path does not go below the x-axis. A peak on a Dyck path is a node that is immediately preceded by a northeast step and immediately followed by a southeast step. A peak is at height k if its y-coordinate is(More)
Let H be the Hankel matrix formed from a sequence of real numbers S = {a 0 = 1, a 1 , a 2 , a 3 , ...}, and let L denote the lower triangular matrix obtained from the Gaussian column reduction of H. This paper gives a matrix-theoretic proof that the associated Stieltjes matrix S L is a tri-diagonal matrix. It is also shown that for any sequence (of nonzero(More)
We consider those lattice paths that use the steps Up, Level , and Down with assigned weights w, u, and v. In probability theory, the total weight is 1. In combi-natorics, we regard weight as the number of colors and normalize by setting w = 1. The lattice paths generate Motzkin sequences. Here we give a combinatorial proof of a three-term recursion for a(More)