Narayana number
Named afterTadepalli Venkata Narayana
No. of known termsinfinity
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).

Formula

The Narayana numbers can be expressed in terms of binomial coefficients:

Numerical values

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

(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 , 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 sub-pattern ():

(()(()))  ((()()))  ((())())
()((()))  (())(())  ((()))()

From this example it should be obvious that , since the only way to get a single sub-pattern () 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:

Monotonic lattice paths

The Narayana numbers also count the number of lattice paths from to , with steps only northeast and southeast, not straying below the x-axis, 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.

Rooted trees

The 6 ordered rooted trees of 4 edges and 2 leaves, corresponding to the Narayana number N(4, 2)
The 6 ordered rooted trees of 4 edges and 2 leaves, corresponding to the Narayana number N(4, 2)

The number of unlabeled ordered rooted trees with edges and leaves is equal to .

This is analogous to the above examples:

Partitions

The 1,6,6,1 non-crossing partitions with 1,2,3,4 blocks of a 4-element set
The 1,6,6,1 non-crossing partitions with 1,2,3,4 blocks of a 4-element set

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 off-topic, 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 non-crossing partitions, we may use the Catalan numbers to count the non-crossing partitions of all elements of the set, . To count the non-crossing partitions in which the set is partitioned in exactly blocks, we use the Narayana number .

Generating function

The generating function for the Narayana numbers is [1]

See also

Citations

  1. ^ Petersen 2015, p. 25.

References