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- Narayana_number abstract "In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana triangle, that occur in various counting problems. They are named for T.V. Narayana (1930–1987), a mathematician from India.The Narayana numbers can be expressed in terms of binomial coefficients: An example of a counting problem whose solution can be given in terms of the Narayana numbers N(n, k), is the number of expressions containing n pairs of parentheses which are correctly matched and which contain k distinct nestings. For instance, N(4, 2) = 6 as with four pairs of parentheses six sequences can be created which each contain two times the sub-pattern '':(()) ()() (()) (()) (()) (())From this example it should be obvious that 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 positions, followed by all the closing parentheses. Also N(n, n) = 1, as 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).The first eight rows of the Narayana triangle read:k = 1 2 3 4 5 6 7 8n = 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 OEIS)The sum of the rows in this triangle equal the Catalan numbers:To illustrate this relationship, the Narayana numbers also count the number of paths from (0, 0) to (2n, 0), with steps only northeast and southeast, not straying below the x-axis, with k peaks.The following figures represent the Narayana numbers N(4, k):The sum of N(4, k) is 1 + 6 + 6 + 1, or 14, which is the same as Catalan number C4. This sum coincides with the interpretation of Catalan numbers as the number of monotonic paths along the edges of an n × n grid that do not pass above the diagonal.".
- Narayana_number thumbnail Narayana41.svg?width=300.
- Narayana_number wikiPageID "12267066".
- Narayana_number wikiPageRevisionID "557574012".
- Narayana_number hasPhotoCollection Narayana_number.
- Narayana_number subject Category:Factorial_and_binomial_topics.
- Narayana_number subject Category:Integer_sequences.
- Narayana_number subject Category:Permutations.
- Narayana_number subject Category:Triangles_of_numbers.
- Narayana_number type Abstraction100002137.
- Narayana_number type Arrangement107938773.
- Narayana_number type Attribute100024264.
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- Narayana_number type Event100029378.
- Narayana_number type Figure113862780.
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- Narayana_number type Happening107283608.
- Narayana_number type IntegerSequences.
- Narayana_number type Ordering108456993.
- Narayana_number type Permutations.
- Narayana_number type PlaneFigure113863186.
- Narayana_number type Polygon113866144.
- Narayana_number type PsychologicalFeature100023100.
- Narayana_number type Sequence108459252.
- Narayana_number type Series108457976.
- Narayana_number type Shape100027807.
- Narayana_number type Substitution107443761.
- Narayana_number type Triangle113879320.
- Narayana_number type TrianglesOfNumbers.
- Narayana_number type Variation107337390.
- Narayana_number type YagoPermanentlyLocatedEntity.
- Narayana_number comment "In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana triangle, that occur in various counting problems. They are named for T.V.".
- Narayana_number label "Narayana number".
- Narayana_number label "Nombre de Narayana".
- Narayana_number sameAs Nombre_de_Narayana.
- Narayana_number sameAs m.02vy858.
- Narayana_number sameAs Q6965520.
- Narayana_number sameAs Q6965520.
- Narayana_number sameAs Narayana_number.
- Narayana_number wasDerivedFrom Narayana_number?oldid=557574012.
- Narayana_number depiction Narayana41.svg.
- Narayana_number isPrimaryTopicOf Narayana_number.