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Date: 26-12-2019
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Date: 29-8-2020
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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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