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A Natural Extension of Catalan Numbers
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Noam Solomon and Shay Solomon

Dept. of Mathematics and Computer Science

Ben-Gurion University of the Negev

Beer-Sheva 84105

Israel

**Abstract:**

A Dyck path is a lattice path in the plane integer lattice **Z**
× **Z** consisting of steps (1,1) and (1,-1), each
connecting diagonal lattice points, which never passes below the
*x*-axis. The number of all Dyck paths that start at (0,0) and finish
at (2*n*,0) is also known as the *n*th Catalan number.
In this paper we find a closed formula, depending on a non-negative
integer *t* and on two lattice points *p*_{1}
and *p*_{2}, for the number
of Dyck paths starting at *p*_{1},
ending at *p*_{2}, and touching the
*x*-axis exactly *t* times.
Moreover, we provide explicit expressions for the corresponding
generating function and bivariate generating function.

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(Concerned with sequence
A000108.)

Received December 30 2007;
revised version received August 7 2008.
Published in *Journal of Integer Sequences*, August 7 2008.
Slight revision, August 17 2008.

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