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Concept
Catalan number
Summary
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan.
The nth Catalan number can be expressed directly in terms of the central binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3, ... are
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... .
An alternative expression for Cn is
for
which is equivalent to the expression given above because . This expression shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula.
Another alternative expression is
which can be directly interpreted in terms of the cycle lemma; see below.
The Catalan numbers satisfy the recurrence relations
and
Asymptotically, the Catalan numbers grow as
in the sense that the quotient of the nth Catalan number and the expression on the right tends towards 1 as n approaches infinity. A more accurate asymptotic analysis shows that the Catalan numbers are approximated by the fourth order approximation.
This can be proved by using the asymptotic growth of the central binomial coefficients, by Stirling's approximation for , or via generating functions.
The only Catalan numbers Cn that are odd are those for which n = 2k − 1; all others are even. The only prime Catalan numbers are C2 = 2 and C3 = 5.
The Catalan numbers have the integral representations
which immediately yields .
This has a simple probabilistic interpretation. Consider a random walk on the integer line, starting at 0. Let -1 be a "trap" state, such that if the walker arrives at -1, it will remain there. The walker can arrive at the trap state at times 1, 3, 5, 7..., and the number of ways the walker can arrive at the trap state at time is . Since the 1D random walk is recurrent, the probability that the walker eventually arrives at -1 is .
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Related concepts (16)
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n.
Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.
Narayana number
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: An example of a counting problem whose solution can be given in terms of the Narayana numbers , is the number of words containing n pairs of parentheses, which are correctly matched (known as Dyck words) and which contain k distinct nestings.