The Narayana numbers *N*(*n*, *k*) describe, among other things, the number of paths
from (0, 0) to (2*n*, 0) that stay above the *x*-axis,
using only steps northeast and southeast, having *k* peaks.

For example, *N*(1, 1) = 1, since there’s only one such path and it has one peak:

Likewise, *N*(2, 1) = 1 and *N*(2, 2) = 1, since there are two paths from
(0, 0) to (4, 0); the first has one peak and the second has two peaks:

Here, *N*(3, 1) = 1;
*N*(3, 2) = 3 [that is, there are three paths that have 2 peaks]; and *N*(3, 3) = 1.

Paths for *N*(4, *k*):

From these, we see that *N*(*n*, 1) = 1 and
*N*(*n*, *n*) = 1.

The totals of *N*(*n*, *k*) as *k* goes from 1 to *n* are
the Catalan numbers, as is apparent from the
path interpretation.

© 1998–2024 by Robert Dickau.

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