The Narayan number
for ,
2, ... and ,
...,
gives a solution to several counting problems in combinatorics. For example, gives the number of expressions
with
pairs of parentheses that are correctly matched and contain distinct nestings. It also gives the number Dyck
paths of length
with exactly
peaks.

MacMahon, P. A. Combinatory Analysis, 2 vols. New York: Chelsea, 1960.Narayana, T. V.
Lattice Path Combinatorics with Statistical Applications. Toronto, Canada:
University of Toronto Press, pp. 100-101, 1979.Stanley, R. P.
Problems 6.36(a) and (b) in Enumerative
Combinatorics, Vol. 2. Cambridge, England: Cambridge University Press,
1999.