Catalan number
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The Catalan numbers are a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the
central binomial coefficient In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first ...
s by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : .


Properties

An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, 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 :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s :C_0 = 1 \quad \text \quad C_=\sum_^C_C_\quad\textn > 0 and :C_0 = 1 \quad \text \quad C_ = \fracC_\quad\textn > 0. Asymptotically, the Catalan numbers grow as C_n \sim \frac\,, in the sense that the quotient of the -th Catalan number and the expression on the right tends towards 1 as approaches infinity. This can be proved by using the asymptotic growth of the central binomial coefficients, by
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related ...
for n!, or via generating functions. The only Catalan numbers that are odd are those for which ; all others are even. The only prime Catalan numbers are and . More generally, the multiplicity with which a prime divides can be determined by first expressing in base . For , the multiplicity is the number of 1 bits, minus 1. For an odd prime, count all digits greater than ; also count digits equal to unless final; and count digits equal to if not final and the next digit is counted. The only known odd Catalan numbers that do not have last digit 5 are , , , , and . The odd Catalan numbers, for , do not have last digit 5 if has a base 5 representation containing 0, 1 and 2 only, except in the least significant place, which could also be a 3. The Catalan numbers have the integral representations :C_n=\frac \int_0^4 x^n\sqrt\,dx\, =\frac4^n\int_^ t^\sqrt\,dt. which immediately yields \sum_^\infty \frac = 2. 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 2k+1 is C_k. Since the 1D random walk is recurrent, the probability that the walker eventually arrives at -1 is \sum_^\infty \frac = 1.


Applications in combinatorics

There are many counting problems in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
whose solution is given by the Catalan numbers. The book ''Enumerative Combinatorics: Volume 2'' by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. Following are some examples, with illustrations of the cases and . * is the number of Dyck words of length . A Dyck word is a
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
consisting of X's and 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 up to length 6:
XY
XXYY     XYXY
XXXYYY     XYXXYY     XYXYXY     XXYYXY     XXYXYY
* Re-interpreting the symbol X as an open
parenthesis A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...
and Y as a close parenthesis, counts the number of expressions containing pairs of parentheses which are correctly matched:
((()))     (()())     (())()     ()(())     ()()()
* is the number of different ways factors can be completely parenthesized (or the number of ways of associating applications of a
binary operator In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation o ...
, as in the matrix chain multiplication problem). For , for example, we have the following five different parenthesizations of four factors:
((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))
* Successive applications of a binary operator can be represented in terms of a full binary tree, by labeling each leaf . It follows that is the number of full binary
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only p ...
with leaves, or, equivalently, with a total of internal nodes: * is the number of non-isomorphic ordered (or plane) trees with vertices. See encoding general trees as binary trees. For example, is the number of possible parse trees for a sentence (assuming binary branching), in natural language processing. * is the number of monotonic
lattice path In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the Set (mathematics), set , is a sequence of Vector (mathematics and physics), vectors such that each consecutive difference v_i - v_ lies in . A l ...
s along the edges of a grid with square cells, which do not pass above the diagonal. A monotonic path is one which starts in the lower left corner, finishes in the upper right corner, and consists entirely of edges pointing rightwards or upwards. Counting such paths is equivalent to counting Dyck words: X stands for "move right" and Y stands for "move up". The following diagrams show the case : This can be represented by listing the Catalan elements by column height:
,0,0,0 ,0,0,1 ,0,0,2 ,0,1,1/div>
,1,1,1 ,0,1,2 ,0,0,3 ,1,1,2 ,0,2,2 ,0,1,3/div>
,0,2,3 ,1,1,3 ,1,2,2 ,1,2,3/div> * A
convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...
with sides can be cut into
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s by connecting vertices with non-crossing
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
s (a form of
polygon triangulation In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may ...
). The number of triangles formed is and the number of different ways that this can be achieved is . The following hexagons illustrate the case : * is the number of
stack Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...
-sortable
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s of . A permutation is called stack-sortable if , where is defined recursively as follows: write where is the largest element in and and are shorter sequences, and set , with being the identity for one-element sequences. * is the number of permutations of that avoid the
permutation pattern In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of entries representing the result of a ...
 123 (or, alternatively, any of the other patterns of length 3); that is, the number of permutations with no three-term increasing subsequence. For , these permutations are 132, 213, 231, 312 and 321. For , they are 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312 and 4321. * is the number of noncrossing partitions of the set . ''A fortiori'', never exceeds the -th
Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of epony ...
. is also the number of noncrossing partitions of the set in which every block is of size 2. * is the number of ways to tile a stairstep shape of height with rectangles. Cutting across the anti-diagonal and looking at only the edges gives full binary trees. The following figure illustrates the case : * is the number of ways to form a "mountain range" with upstrokes and downstrokes that all stay above a horizontal line. The mountain range interpretation is that the mountains will never go below the horizon.
* is the number of standard Young tableaux whose diagram is a 2-by- rectangle. In other words, it is the number of ways the numbers can be arranged in a 2-by- rectangle so that each row and each column is increasing. As such, the formula can be derived as a special case of the hook-length formula.
123   124   125   134   135
456   356   346   256   246
* C_n is the number of length sequences that start with 1, and can increase by either 0 or 1, or decrease by any number (to at least 1). For n=4 these are 1234, 1233, 1232, 1231, 1223, 1222, 1221, 1212, 1211, 1123, 1122, 1121, 1112, 1111. From a Dyck path, start a counter at . An X increases the counter by and a Y decreases it by . Record the values at only the X's. Compared to the similar representation of the Bell numbers, only 1213 is missing.


Proof of the formula

There are several ways of explaining why the formula :C_n = \frac solves the combinatorial problems listed above. The first proof below uses a
generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
. The other proofs are examples of
bijective proof In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other ...
s; they involve literally counting a collection of some kind of object to arrive at the correct formula.


First proof

We first observe that all of the combinatorial problems listed above satisfy Segner's
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:C_0 = 1 \quad \text \quad C_=\sum_^n C_i\,C_\quad\textn\ge 0. For example, every Dyck word of length ≥ 2 can be written in a unique way in the form : with (possibly empty) Dyck words and . The
generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
for the Catalan numbers is defined by :c(x)=\sum_^\infty C_n x^n. The recurrence relation given above can then be summarized in generating function form by the relation :c(x)=1+xc(x)^2; in other words, this equation follows from the recurrence relation by expanding both sides into
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
. On the one hand, the recurrence relation uniquely determines the Catalan numbers; on the other hand, interpreting as a
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
of and using the
quadratic formula In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...
, the generating function relation can be algebraically solved to yield two solution possibilities :c(x) = \frac  or  c(x) = \frac. From the two possibilities, the second must be chosen because only the second gives :C_0 = \lim_ c(x) = 1. The square root term can be expanded as a power series using the
binomial series In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where \alpha is any complex number, and the power series on the right-hand side is expressed in terms of the ...
\begin 1 - \sqrt & = -\sum_^ \binom(-4x)^ = -\sum_^ \frac(-4x)^ \\ &= -\sum_^ \frac(-4x)^ = \sum_^ \fracx^ \\ & = \sum_^ \fracx^ = \sum_^ \frac \binomx^\,. \end Thus, c(x) = \frac = \sum_^ \frac \binomx^\,.


Second proof

We count the number of paths which start and end on the diagonal of an grid. All such paths have right and up steps. Since we can choose which of the steps are up or right, there are in total \tbinom monotonic paths of this type. A ''bad'' path crosses the main diagonal and touches the next higher diagonal (red in the illustration). The part of the path after the higher diagonal is then flipped about that diagonal, as illustrated with the red dotted line. This swaps all the right steps to up steps and vice versa. In the section of the path that is not reflected, there is one more up step than right steps, so therefore the remaining section of the bad path has one more right step than up steps. When this portion of the path is reflected, it will have one more up step than right steps. Since there are still steps, there are now up steps and right steps. So, instead of reaching , all bad paths after reflection end at . Because every monotonic path in the grid meets the higher diagonal, and because the reflection process is reversible, the reflection is therefore a bijection between bad paths in the original grid and monotonic paths in the new grid. The number of bad paths is therefore: : = = and the number of Catalan paths (i.e. good paths) is obtained by removing the number of bad paths from the total number of monotonic paths of the original grid, :C_n = - = \frac. In terms of Dyck words, we start with a (non-Dyck) sequence of X's and Y's and interchange all X's and Y's after the first Y that violates the Dyck condition. After this Y, note that there is exactly one more Y than there are Xs.


Third proof

This bijective proof provides a natural explanation for the term appearing in the denominator of the formula for . A generalized version of this proof can be found in a paper of Rukavicka Josef (2011). Given a monotonic path, the exceedance of the path is defined to be the number of vertical edges above the diagonal. For example, in Figure 2, the edges above the diagonal are marked in red, so the exceedance of this path is 5. Given a monotonic path whose exceedance is not zero, we apply the following algorithm to construct a new path whose exceedance is less than the one we started with. * Starting from the bottom left, follow the path until it first travels above the diagonal. * Continue to follow the path until it ''touches'' the diagonal again. Denote by the first such edge that is reached. * Swap the portion of the path occurring before with the portion occurring after . In Figure 3, the black dot indicates the point where the path first crosses the diagonal. The black edge is , and we place the last lattice point of the red portion in the top-right corner, and the first lattice point of the green portion in the bottom-left corner, and place X accordingly, to make a new path, shown in the second diagram. The exceedance has dropped from to . In fact, the algorithm causes the exceedance to decrease by for any path that we feed it, because the first vertical step starting on the diagonal (at the point marked with a black dot) is the only vertical edge that changes from being above the diagonal to being below it when we apply the algorithm - all the other vertical edges stay on the same side of the diagonal. It can be seen that this process is ''reversible'': given any path whose exceedance is less than , there is exactly one path which yields when the algorithm is applied to it. Indeed, the (black) edge , which originally was the first horizontal step ending on the diagonal, has become the ''last'' horizontal step ''starting'' on the diagonal. Alternatively, reverse the original algorithm to look for the first edge that passes ''below'' the diagonal. This implies that the number of paths of exceedance is equal to the number of paths of exceedance , which is equal to the number of paths of exceedance , and so on, down to zero. In other words, we have split up the set of ''all'' monotonic paths into equally sized classes, corresponding to the possible exceedances between 0 and . Since there are \textstyle monotonic paths, we obtain the desired formula \textstyle C_n = \frac. Figure 4 illustrates the situation for . Each of the 20 possible monotonic paths appears somewhere in the table. The first column shows all paths of exceedance three, which lie entirely above the diagonal. The columns to the right show the result of successive applications of the algorithm, with the exceedance decreasing one unit at a time. There are five rows, that is , and the last column displays all paths no higher than the diagonal. Using Dyck words, start with a sequence from \textstyle \binom. Let X_d be the first that brings an initial subsequence to equality, and configure the sequence as (F)X_d(L). The new sequence is LXF.


Fourth proof

This proof uses the triangulation definition of Catalan numbers to establish a relation between and . Given a polygon with sides and a triangulation, mark one of its sides as the base, and also orient one of its total edges. There are such marked triangulations for a given base. Given a polygon with sides and a (different) triangulation, again mark one of its sides as the base. Mark one of the sides other than the base side (and not an inner triangle edge). There are such marked triangulations for a given base. There is a simple bijection between these two marked triangulations: We can either collapse the triangle in whose side is marked (in two ways, and subtract the two that cannot collapse the base), or, in reverse, expand the oriented edge in to a triangle and mark its new side. Thus :(4n+2)C_n = (n+2)C_. Write \textstyle\fracC_ = C_n. Because :(2n)!=(2n)!!(2n-1)!!=2^nn!(2n-1)!! we have :\frac=2^n(2n-1)!!=(4n-2)!!!!. Applying the recursion with C_0=1 gives the result.


Fifth proof

This proof is based on the Dyck words interpretation of the Catalan numbers, so C_n is the number of ways to correctly match pairs of brackets. We denote a (possibly empty) correct string with and its inverse with . Since any can be uniquely decomposed into c = (c_1) c_2, summing over the possible lengths of c_1 immediately gives the recursive definition :C_0 = 1 \quad \text \quad C_ = \sum_^n C_i\,C_\quad\textn\ge 0. Let be a balanced string of length , i.e. contains an equal number of ( and ), so \textstyle B_n = . A balanced string can also be uniquely decomposed into either (c)b or )c'(b, so :B_ = 2\sum_^n B_i C_. Any incorrect (non-Catalan) balanced string starts with c), and the remaining string has one more ( than ), so :B_ - C_ = \sum_^n C_ Also, from the definitions, we have: :B_ - C_ = 2\sum_^n B_i C_ - \sum_^n C_i\,C_ = \sum_^n (2B_i-C_i) C_. Therefore, as this is true for all , :2B_i - C_i = \binom :C_i = 2B_i - \binom :C_i = 2\binom - \binom :C_i=\frac\binom


Sixth proof

This proof is based on the Dyck words interpretation of the Catalan numbers and uses the cycle lemma of Dvoretzky and Motzkin. We call a sequence of X's and Y's ''dominating'' if, reading from left to right, the number of X's is always strictly greater than the number of Y's. The cycle lemma states that any sequence of m X's and n Y's, where m> n, has precisely m-n dominating
circular shift In combinatorial mathematics, a circular shift is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse ope ...
s. To see this, arrange the given sequence of m+n X's and Y's in a circle. Repeatedly removing XY pairs leaves exactly m-n X's. Each of these X's was the start of a dominating circular shift before anything was removed. For example, consider \mathit. This sequence is dominating, but none of its circular shifts \mathit, \mathit, \mathit and \mathit are. A string is a Dyck word of n X's and n Y's if and only if prepending an X to the Dyck word gives a dominating sequence with n+1 X's and n Y's, so we can count the former by instead counting the latter. In particular, when m=n+1, there is exactly one dominating circular shift. There are \textstyle sequences with exactly n+1 X's and n Y's. For each of these, only one of the 2n+1 circular shifts is dominating. Therefore there are \textstyle\frac=C_n distinct sequences of n+1 X's and n Y's that are dominating, each of which corresponds to exactly one Dyck word.


Hankel matrix

The
Hankel matrix In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example, \qquad\begin a & b & c & d & e \\ b & c & d & e & ...
whose entry is the Catalan number has
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1, regardless of the value of . For example, for we have :\det\begin1 & 1 & 2 & 5 \\ 1 & 2 & 5 & 14 \\ 2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132\end = 1. Moreover, if the indexing is "shifted" so that the entry is filled with the Catalan number then the determinant is still 1, regardless of the value of . For example, for we have :\det\begin1 & 2 & 5 & 14 \\ 2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132 \\ 14 & 42 & 132 & 429 \end = 1. Taken together, these two conditions uniquely define the Catalan numbers. Another feature unique to the Catalan–Hankel matrix is that the submatrix starting at has determinant . :\det\begin 2 \end = 2 :\det\begin 2 & 5 \\5 & 14 \end = 3 :\det\begin 2 & 5 & 14\\5 & 14 & 42\\ 14 & 42 & 132\end = 4 :\det\begin 2 & 5 & 14 & 42 \\ 5 & 14 & 42 & 132 \\ 14 & 42 & 132 & 42 9\\ 42 & 132 & 429 & 1430\end = 5 et cetera.


History

The Catalan sequence was described in 1751 by
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, 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 during his exploration of the Towers of Hanoi puzzle. The reflection counting trick (second proof) for Dyck words was found by Désiré André in 1887. The name “Catalan numbers” originated from John Riordan. In 1988, it came to light that the Catalan number sequence had been used in
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
by the Mongolian mathematician Mingantu by 1730. That is when he started to write his book ''Ge Yuan Mi Lu Jie Fa'' '' he Quick Method for Obtaining the Precise Ratio of Division of a Circle', which was completed by his student Chen Jixin in 1774 but published sixty years later. Peter J. Larcombe (1999) sketched some of the features of the work of Mingantu, including the stimulus of Pierre Jartoux, who brought three infinite series to China early in the 1700s. For instance, Ming used the Catalan sequence to express series expansions of \sin(2 \alpha) and \sin(4 \alpha) in terms of \sin(\alpha).


Generalizations

The Catalan numbers can be interpreted as a special case of the
Bertrand's ballot theorem In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives ''p'' votes and candidate B receives ''q'' votes with ''p'' > ''q'', what is the probability that A will be strictly ahead of B throug ...
. Specifically, C_n is the number of ways for a candidate A with votes to lead candidate B with votes. The two-parameter sequence of non-negative integers \frac is a generalization of the Catalan numbers. These are named super-Catalan numbers, per Ira Gessel. These should not confused with the Schröder–Hipparchus numbers, which sometimes are also called super-Catalan numbers. For m=1, this is just two times the ordinary Catalan numbers, and for m=n, the numbers have an easy combinatorial description. However, other combinatorial descriptions are only known for m=2, 3 and 4, and it is an open problem to find a general combinatorial interpretation. Sergey Fomin and Nathan Reading have given a generalized Catalan number associated to any finite crystallographic
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
, namely the number of fully commutative elements of the group; in terms of the associated
root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
, it is the number of anti-chains (or order ideals) in the poset of positive roots. The classical Catalan number C_n corresponds to the root system of type A_n. The classical recurrence relation generalizes: the Catalan number of a Coxeter diagram is equal to the sum of the Catalan numbers of all its maximal proper sub-diagrams. The Catalan numbers are a solution of a version of the
Hausdorff moment problem In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :m_n = \int_0^1 x^n\,d\mu(x) of some Borel measure supported on the clos ...
. For coprime positive integers and , the ''rational Catalan numbers'' \frac \binom count the number of lattice paths with steps of unit length rightwards and upwards from to that never go above the line .


Catalan k-fold convolution

The Catalan -fold convolution, where , is: : \sum_ C_\cdots C_ = \begin \dfracC_, & m \text\\ pt \dfracC_, & m \text \end


See also

* Associahedron *
Bertrand's ballot theorem In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives ''p'' votes and candidate B receives ''q'' votes with ''p'' > ''q'', what is the probability that A will be strictly ahead of B throug ...
*
Binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to ...
* Catalan's triangle * Catalan–Mersenne number *
Delannoy number In mathematics, a Delannoy number D counts the paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (''m'', ''n''), using only single steps north, northeast, or east. The Delannoy numbers are named after French army ...
* Fuss–Catalan number * List of factorial and binomial topics * Lobb numbers * Motzkin number *
Narayana number In combinatorics, the Narayana numbers \operatorname(n, k), n \in \mathbb^+, 1 \le k \le n form a triangular array of natural numbers, called the Narayana triangle, that occur in various Combinatorial enumeration, counting problems. They are named ...
* Narayana polynomials *
Schröder number In mathematics, the Schröder number S_n, also called a ''large Schröder number'' or ''big Schröder number'', describes the number of lattice paths from the southwest corner (0,0) of an n \times n grid to the northeast corner (n,n), using only ...
* Schröder–Hipparchus number *
Semiorder In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given margin of error are deemed incompar ...
* Tamari lattice * Wedderburn–Etherington number * Wigner's semicircle law


Notes


References

* Stanley, Richard P. (2015), ''Catalan numbers''. Cambridge University Press, . *
Conway Conway may refer to: Places United States * Conway, Arkansas * Conway County, Arkansas * Lake Conway, Arkansas * Conway, Florida * Conway, Iowa * Conway, Kansas * Conway, Louisiana * Conway, Massachusetts * Conway, Michigan * Conway Townshi ...
and Guy (1996) '' The Book of Numbers''. New York: Copernicus, pp. 96–106. * * * Koshy, Thomas & Zhenguang Gao (2011) "Some divisibility properties of Catalan numbers",
Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...
95:96–102. * * * *


External links

* * * Davis, Tom
Catalan numbers
Still more examples. * "Equivalence of Three Catalan Number Interpretations" from The Wolfram Demonstrations Projec

* {{DEFAULTSORT:Catalan Number Integer sequences Factorial and binomial topics Enumerative combinatorics Eponymous numbers in mathematics Articles containing proofs