In combinatorics, the Narayana numbers N ( n , k ) , n ∈ N + , 1 ≤ k ≤ n {\displaystyle \operatorname {N} (n,k),n\in \mathbb {N} ^{+},1\leq k\leq n} form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T. V. Narayana (1930–1987).
Formula The Narayana numbers can be expressed in terms of binomial coefficients:
N ( n , k ) = 1 k ( n − 1 k − 1 ) ( n k − 1 ) = 1 n ( n k ) ( n k − 1 ) {\displaystyle \operatorname {N} (n,k)={\frac {1}{k}}{n-1 \choose k-1}{n \choose k-1}={\frac {1}{n}}{n \choose k}{n \choose k-1}}
Numerical values The first eight rows of the Narayana triangle read:
(sequence A001263 in the OEIS)
Combinatorial interpretations
Dyck words An example of a counting problem whose solution can be given in terms of the Narayana numbers N ( n , k ) {\displaystyle \operatorname {N} (n,k)} , is the number of words containing n {\displaystyle n} pairs of parentheses, which are correctly matched (known as Dyck words) and which contain k {\displaystyle k} distinct nestings. For instance, N ( 4 , 2 ) = 6 {\displaystyle \operatorname {N} (4,2)=6} , since with four pairs of parentheses, six sequences can be created which each contain two occurrences the sub-pattern ():
(()(())) ((()())) ((())()) ()((())) (())(()) ((()))()
From this example it should be obvious that N ( n , 1 ) = 1 {\displaystyle \operatorname {N} (n,1)=1} , since the only way to get a single sub-pattern () is to have all the opening parentheses in the first n {\displaystyle n} positions, followed by all the closing parentheses. Also N ( n , n ) = 1 {\displaystyle \operatorname {N} (n,n)=1} , as n {\displaystyle n} distinct nestings can be achieved only by the repetitive pattern ()()()…(). More generally, it can be shown that the Narayana triangle is symmetric:
N ( n , k ) = N ( n , n − k + 1 ) {\displaystyle \operatorname {N} (n,k)=\operatorname {N} (n,n-k+1)}
The sum of the rows in this triangle equal the Catalan numbers:
N ( n , 1 ) + N ( n , 2 ) + N ( n , 3 ) + ⋯ + N ( n , n ) = C n {\displaystyle \operatorname {N} (n,1)+\operatorname {N} (n,2)+\operatorname {N} (n,3)+\cdots +\operatorname {N} (n,n)=C_{n}}
Monotonic lattice paths The Narayana numbers also count the number of lattice paths from ( 0 , 0 ) {\displaystyle (0,0)} to ( 2 n , 0 ) {\displaystyle (2n,0)} , with steps only northeast and southeast, not straying below the x-axis, with k {\displaystyle k} peaks. The following figures represent the Narayana numbers N ( 4 , k ) {\displaystyle \operatorname {N} (4,k)} , illustrating the above-mentioned symmetries.
The sum of N ( 4 , k ) {\displaystyle \operatorname {N} (4,k)} is 1 + 6 + 6 + 1 = 14, which is the 4th Catalan number, C 4 {\displaystyle C_{4}} . This sum coincides with the interpretation of Catalan numbers as the number of monotonic paths along the edges of an n × n {\displaystyle n\times n} grid that do not pass above the diagonal.
Rooted trees
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