Named after  Tadepalli Venkata Narayana 

No. of known terms  infinity 
Formula  
OEIS index 

In combinatorics, the Narayana numbers 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).
The Narayana numbers can be expressed in terms of binomial coefficients:
The first eight rows of the Narayana triangle read:
n  k  

1  2  3  4  5  6  7  8  
1  1  
2  1  1  
3  1  3  1  
4  1  6  6  1  
5  1  10  20  10  1  
6  1  15  50  50  15  1  
7  1  21  105  175  105  21  1  
8  1  28  196  490  490  196  28  1 
An example of a counting problem whose solution can be given in terms of the Narayana numbers , is the number of words containing pairs of parentheses, which are correctly matched (known as Dyck words) and which contain distinct nestings. For instance, , since with four pairs of parentheses, six sequences can be created which each contain two occurrences the subpattern ()
:
(()(())) ((()())) ((())()) ()((())) (())(()) ((()))()
From this example it should be obvious that , since the only way to get a single subpattern ()
is to have all the opening parentheses in the first positions, followed by all the closing parentheses. Also , as distinct nestings can be achieved only by the repetitive pattern ()()()…()
.
More generally, it can be shown that the Narayana triangle is symmetric:
The sum of the rows in this triangle equal the Catalan numbers:
The Narayana numbers also count the number of lattice paths from to , with steps only northeast and southeast, not straying below the xaxis, with peaks.
The following figures represent the Narayana numbers , illustrating the above mentioned symmetries.
Paths  

N(4, 1) = 1 path with 1 peak  
N(4, 2) = 6 paths with 2 peaks:  
N(4, 3) = 6 paths with 3 peaks:  
N(4, 4) = 1 path with 4 peaks: 
The sum of is 1 + 6 + 6 + 1 = 14, which is the 4th Catalan number, . This sum coincides with the interpretation of Catalan numbers as the number of monotonic paths along the edges of an grid that do not pass above the diagonal.
The number of unlabeled ordered rooted trees with edges and leaves is equal to .
This is analogous to the above examples:
()
. In analogous fashion, we can construct a Dyck word from a rooted tree via a depthfirst search. Thus, there is an isomorphism between Dyck words and rooted trees.In the study of partitions, we see that in a set containing elements, we may partition that set in different ways, where is the ^{th} Bell number. Furthermore, the number of ways to partition a set into exactly blocks we use the Stirling numbers . Both of these concepts are a bit offtopic, but a necessary foundation for understanding the use of the Narayana numbers. In both of the above two notions crossing partitions are accounted for.
To reject the crossing partitions and count only the noncrossing partitions, we may use the Catalan numbers to count the noncrossing partitions of all elements of the set, . To count the noncrossing partitions in which the set is partitioned in exactly blocks, we use the Narayana number .
The generating function for the Narayana numbers is ^{[1]}