Narayana Number
المؤلف:
MacMahon, P. A.
المصدر:
Combinatory Analysis, 2 vols. New York: Chelsea, 1960.
الجزء والصفحة:
...
5-1-2021
1448
Narayana Number
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.
A closed-form expression of 
is given by
where 
is a binomial coefficient.
Summing over 
gives the Catalan number
Enumerating 
as a number triangle is called the Narayana triangle.
REFERENCES:
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.
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