Catalan number
The Catalan numbers, named after the Belgian mathematician Eugène Charles Catalan (1814—1894), form a sequence of natural numbers that occur in various counting problems in combinatorics. The nth Catalan number is defined by
(see binomial coefficient). The first Catalan numbers for n = 0, 1, 2, 3, ... are - 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452...
All Catalan numbers are natural numbers because one can verify that -
Applications in combinatorics
- Cn is the number of Dyck words of length 2n. A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words of length 6: XXXYYY, XYXXYY, XYXYXY, XXYYXY, XXYXYY. Accordingly, C3 = 5. Thinking of X as an open parenthesis and of Y as a closed one, every Dyck word of length 2n can be seen as an expression with n pairs of parentheses correctly placed together: ((())), ()(()), ()()(), (())(), (()()). Dyck words can be naturally represented as certain paths in a grid with n+1 by n+1 vertices connected by vertical and horizontal lines. The paths start in the lower left corner, end in the upper right corner, always move up or right, and never pass above the diagonal. X stands for "move right" and Y stands for "move up".
One can count Dyck words with the following trick due to D. André: focus on those words containing n X's and n Y's that are not Dyck words. In such a word, find the first Y that violates the Dyck condition, and then flip all letters following this Y from Y to X and vice versa. You obtain a word with n + 1 Y's and n − 1 X's, and in fact every word with n + 1 Y's and n − 1 X's can be obtained in this fashion in precisely one way. The number of these words is equal to
which is therefore the same as the number of non-Dyck words; the number of Dyck words must then be
which is the nth Catalan number Cn. - Cn is the number of different ways n + 1 factors can be completely parenthesized. For n = 3 for example, we have the following 5 different parenthesizations of 4 factors: a(b(cd)), a((bc)d), (ab)(cd), (a(bc))d, ((ab)c)d. Such expressions can be naturally represented as rooted ordered binary treess, so Cn also counts the number of these trees with n + 1 leaves.
- Cn is the number of different ways a polygon with n + 2 sides can be cut into triangless by connecting vertices with straight lines.
If w is a finite sequence of different integers, we define a reordering S(w) of w recursively as follows: write w = unv where n is the largest element in w and u and v are shorter sequences, and set S(w) = S(u)S(v)n, with S being the identity for one-element sequences. A permutation w of {1, ..., n} is called \'stack-sortable' if S(w) = (1, ..., n). The number of stack-sortable permutations of {1, ..., n} is equal to Cn. - Cn is the number of noncrossing partitions of the set { 1, ..., n }. A fortiori, Cn never exceeds the nth Bell number. Cn is also the number of noncrossing partitions of the set { 1, ..., 2n } in which every block is of size 2. The conjunction of these two facts may be used in a proof by mathematical induction that all of the free cumulants of degree more than 2 of the Wigner semicircle law are zero.
- Cn is the number of non-isomorphic full binary trees with n vertices that have children, usually called internal vertices or branches. (A rooted binary tree is full if every vertex has either two children or no children.)
Occurrence in probability theory
The Catalan numbers occur in a simple expression for the moments of the Wigner semicircle distribution, which is important in free probability theory and the theory of random matrices.
Recurrence relation and asymptotic expression The Catalan numbers satisfy the recurrence relation
: This follows from the fact that every Dyck word w of length ≥ 2 can be written in a unique way in the form - w = Xw1Yw2
with (possibly empty) Dyck words w1 and w2. The generating function for the Catalan numbers is defined by
and using the above recurrence relation we see that
and hence
Asymptotically, the Catalan numbers grow as
in the sense that the quotient of the n-th Catalan number and the expression on the right tends towards 1 for
n → ∞. (This can be proved by using Stirling's approximation for n!.)
History The Catalan sequence was first described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions. The counting trick for Dyck words was found by D. André in 1887. It has been shown that the number of possible interpretations of a sentence, function of the number of prepositional groups ("he saw a man on the hill with a telescope"), is the Catalan number series.
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