Catalan number

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For names of numbers in Catalan, see List of numbers in various languages § Occitano-Romance. Catalan number_sentence_0

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. Catalan number_sentence_1

They are named after the Belgian mathematician Eugène Charles Catalan (1814–1894). Catalan number_sentence_2

The nth Catalan number is given directly in terms of binomial coefficients by Catalan number_sentence_3

The first Catalan numbers for n = 0, 1, 2, 3, ... are Catalan number_sentence_4

Catalan number_description_list_0

  • 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, ... (sequence in the OEIS).Catalan number_item_0_0

Properties Catalan number_section_0

An alternative expression for Cn is Catalan number_sentence_5

The Catalan numbers satisfy the recurrence relations Catalan number_sentence_6

and Catalan number_sentence_7

Asymptotically, the Catalan numbers grow as Catalan number_sentence_8

in the sense that the quotient of the nth Catalan number and the expression on the right tends towards 1 as n approaches infinity. Catalan number_sentence_9

This can be proved by using Stirling's approximation for n! Catalan number_sentence_10

or via generating functions. Catalan number_sentence_11

The only Catalan numbers Cn that are odd are those for which n = 2 − 1; all others are even. Catalan number_sentence_12

The only prime Catalan numbers are C2 = 2 and C3 = 5. Catalan number_sentence_13

The Catalan numbers have an integral representation Catalan number_sentence_14

Applications in combinatorics Catalan number_section_1

There are many counting problems in combinatorics whose solution is given by the Catalan numbers. Catalan number_sentence_15

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. Catalan number_sentence_16

Following are some examples, with illustrations of the cases C3 = 5 and C4 = 14. Catalan number_sentence_17

Catalan number_unordered_list_1

  • 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:Catalan number_item_1_1

Catalan number_unordered_list_2

  • Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, Cn counts the number of expressions containing n pairs of parentheses which are correctly matched:Catalan number_item_2_2

Catalan number_unordered_list_3

  • Cn is the number of different ways n + 1 factors can be completely parenthesized (or the number of ways of associating n applications of a binary operator). For n = 3, for example, we have the following five different parenthesizations of four factors:Catalan number_item_3_3

Catalan number_unordered_list_4

  • Successive applications of a binary operator can be represented in terms of a full binary tree. (A rooted binary tree is full if every vertex has either two children or no children.) It follows that Cn is the number of full binary trees with n + 1 leaves:Catalan number_item_4_4

Catalan number_unordered_list_5

  • Cn is the number of non-isomorphic ordered (or plane) trees with n + 1 vertices.Catalan number_item_5_5
  • Cn is the number of monotonic lattice paths along the edges of a grid with n × n 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".Catalan number_item_5_6

The following diagrams show the case n = 4: Catalan number_sentence_18

This can be succinctly represented by listing the Catalan elements by column height: Catalan number_sentence_19

Catalan number_unordered_list_6

  • A convex polygon with n + 2 sides can be cut into triangles by connecting vertices with non-crossing line segments (a form of polygon triangulation). The number of triangles formed is n and the number of different ways that this can be achieved is Cn. The following hexagons illustrate the case n = 4:Catalan number_item_6_7

Catalan number_unordered_list_7

  • Cn is the number of stack-sortable permutations of {1, ..., n}. A permutation w is called stack-sortable if S(w) = (1, ..., n), where S(w) is defined 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.Catalan number_item_7_8
  • Cn is the number of permutations of {1, ..., n} that avoid the permutation pattern 123 (or, alternatively, any of the other patterns of length 3); that is, the number of permutations with no three-term increasing subsequence. For n = 3, these permutations are 132, 213, 231, 312 and 321. For n = 4, they are 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312 and 4321.Catalan number_item_7_9
  • 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. This law is important in free probability theory and the theory of random matrices.Catalan number_item_7_10
  • Cn is the number of ways to tile a stairstep shape of height n with n rectangles. The following figure illustrates the case n = 4:Catalan number_item_7_11

Catalan number_unordered_list_8

  • Cn is the number of ways to form a "mountain range" with n upstrokes and n downstrokes that all stay above a horizontal line. The mountain range interpretation is that the mountains will never go below the horizon.Catalan number_item_8_12

Catalan number_unordered_list_9

  • Cn is the number of standard Young tableaux whose diagram is a 2-by-n rectangle. In other words, it is the number of ways the numbers 1, 2, ..., 2n can be arranged in a 2-by-n 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.Catalan number_item_9_13
  • Cn is the number of ways that the vertices of a convex 2n-gon can be paired so that the line segments joining paired vertices do not intersect. This is precisely the condition that guarantees that the paired edges can be identified (sewn together) to form a closed surface of genus zero (a topological 2-sphere).Catalan number_item_9_14
  • Cn is the number of semiorders on n unlabeled items.Catalan number_item_9_15
  • In chemical engineering Cn−1 is the number of possible separation sequences which can separate a mixture of n components.Catalan number_item_9_16

Proof of the formula Catalan number_section_2

There are several ways of explaining why the formula Catalan number_sentence_20

solves the combinatorial problems listed above. Catalan number_sentence_21

The first proof below uses a generating function. Catalan number_sentence_22

The other proofs are examples of bijective proofs; they involve literally counting a collection of some kind of object to arrive at the correct formula. Catalan number_sentence_23

First proof Catalan number_section_3

We first observe that all of the combinatorial problems listed above satisfy Segner's recurrence relation Catalan number_sentence_24

For example, every Dyck word w of length ≥ 2 can be written in a unique way in the form Catalan number_sentence_25

Catalan number_description_list_10

  • w = Xw1Yw2Catalan number_item_10_17

with (possibly empty) Dyck words w1 and w2. Catalan number_sentence_26

The generating function for the Catalan numbers is defined by Catalan number_sentence_27

The recurrence relation given above can then be summarized in generating function form by the relation Catalan number_sentence_28

in other words, this equation follows from the recurrence relation by expanding both sides into power series. Catalan number_sentence_29

On the one hand, the recurrence relation uniquely determines the Catalan numbers; on the other hand, the generating function relation can be algebraically solved to yield Catalan number_sentence_30

Choosing the minus sign (in the first expression), the fraction has a power series at 0 so its coefficients must therefore be the Catalan numbers. Catalan number_sentence_31

This solution satisfies Catalan number_sentence_32

The other solution, with the plus sign, has a pole at 0 so it cannot be a valid solution for c(x). Catalan number_sentence_33

The square root term can be expanded as a power series using the identity Catalan number_sentence_34

This is a special case of Newton's generalized binomial theorem; as with the general theorem, it can be proved by computing derivatives to produce its Taylor series. Catalan number_sentence_35

Setting y = −4x and substituting this power series into the expression for c(x) and shifting the summation index n by 1, the expansion simplifies to Catalan number_sentence_36

The coefficients are now the desired formula for Cn. Catalan number_sentence_37

Second proof Catalan number_section_4

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, Catalan number_sentence_38

In terms of Dyck words, we start with a (non-Dyck) sequence of n X's and n Y's and interchange all X's and Y's after the first Y that violates the Dyck condition. Catalan number_sentence_39

At that first Y, there are k + 1 Y's and k X's for some k between 1 and n − 1. Catalan number_sentence_40

Third proof Catalan number_section_5

The following bijective proof, while being more involved than the previous one, provides a more natural explanation for the term n + 1 appearing in the denominator of the formula for Cn. Catalan number_sentence_41

A generalized version of this proof can be found in a paper of Rukavicka Josef (2011). Catalan number_sentence_42

Suppose we are given a monotonic path, which may happen to cross the diagonal. Catalan number_sentence_43

The exceedance of the path is defined to be the number of vertical edges which lie above the diagonal. Catalan number_sentence_44

For example, in Figure 2, the edges lying above the diagonal are marked in red, so the exceedance of the path is 5. Catalan number_sentence_45

Now, if we are given a monotonic path whose exceedance is not zero, then we may apply the following algorithm to construct a new path whose exceedance is one less than the one we started with. Catalan number_sentence_46

Catalan number_unordered_list_11

  • Starting from the bottom left, follow the path until it first travels above the diagonal.Catalan number_item_11_18
  • Continue to follow the path until it touches the diagonal again. Denote by X the first such edge that is reached.Catalan number_item_11_19
  • Swap the portion of the path occurring before X with the portion occurring after X.Catalan number_item_11_20

The following example should make this clearer. Catalan number_sentence_47

In Figure 3, the black dot indicates the point where the path first crosses the diagonal. Catalan number_sentence_48

The black edge is X, and we swap the red portion with the green portion to make a new path, shown in the second diagram. Catalan number_sentence_49

The exceedance has dropped from three to two. Catalan number_sentence_50

In fact, the algorithm will cause the exceedance to decrease by one, 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 unique vertical edge that under the operation passes from above the diagonal to below it; all other vertical edges stay on the same side of the diagonal. Catalan number_sentence_51

It is also not difficult to see that this process is reversible: given any path P whose exceedance is less than n, there is exactly one path which yields P when the algorithm is applied to it. Catalan number_sentence_52

Indeed, the (black) edge X, which originally was the first horizontal step ending on the diagonal, has become the last horizontal step starting on the diagonal. Catalan number_sentence_53

This implies that the number of paths of exceedance n is equal to the number of paths of exceedance n − 1, which is equal to the number of paths of exceedance n − 2, and so on, down to zero. Catalan number_sentence_54

In other words, we have split up the set of all monotonic paths into n + 1 equally sized classes, corresponding to the possible exceedances between 0 and n. Since there are Catalan number_sentence_55

monotonic paths, we obtain the desired formula Catalan number_sentence_56

Figure 4 illustrates the situation for n = 3. Catalan number_sentence_57

Each of the 20 possible monotonic paths appears somewhere in the table. Catalan number_sentence_58

The first column shows all paths of exceedance three, which lie entirely above the diagonal. Catalan number_sentence_59

The columns to the right show the result of successive applications of the algorithm, with the exceedance decreasing one unit at a time. Catalan number_sentence_60

There are five rows, that is, C3 = 5. Catalan number_sentence_61

Fourth proof Catalan number_section_6

This proof uses the triangulation definition of Catalan numbers to establish a relation between Cn and Cn+1. Catalan number_sentence_62

Given a polygon P with n + 2 sides, first mark one of its sides as the base. Catalan number_sentence_63

If P is then triangulated, we can further choose and orient one of its 2n + 1 edges. Catalan number_sentence_64

There are (4n + 2)Cn such decorated triangulations. Catalan number_sentence_65

Now given a polygon Q with n + 3 sides, again mark one of its sides as the base. Catalan number_sentence_66

If Q is triangulated, we can further mark one of the sides other than the base side. Catalan number_sentence_67

There are (n + 2)Cn + 1 such decorated triangulations. Catalan number_sentence_68

Then there is a simple bijection between these two kinds of decorated triangulations: We can either collapse the triangle in Q whose side is marked, or in reverse expand the oriented edge in P to a triangle and mark its new side. Catalan number_sentence_69

Thus Catalan number_sentence_70

The binomial formula for Cn follows immediately from this relation and the initial condition C1 = 1. Catalan number_sentence_71

Fifth proof Catalan number_section_7

This proof is based on the Dyck words interpretation of the Catalan numbers, so Cn is the number of ways to correctly match n pairs of brackets. Catalan number_sentence_72

We denote a (possibly empty) correct string with c and its inverse (where "[" and "]" are exchanged) with c. Since any c can be uniquely decomposed into c = [ c1 ] c2, summing over the possible spots to place the closing bracket immediately gives the recursive definition Catalan number_sentence_73

Also, any incorrect balanced string starts with c ], so Catalan number_sentence_74

Subtracting the above equations and using Bi = di Ci gives Catalan number_sentence_75

Comparing coefficients with the original recursion formula for Cn gives di = i + 1, so Catalan number_sentence_76

Sixth proof Catalan number_section_8

Hankel matrix Catalan number_section_9

The n×n Hankel matrix whose (i, j) entry is the Catalan number Ci+j−2 has determinant 1, regardless of the value of n. For example, for n = 4 we have Catalan number_sentence_77

Moreover, if the indexing is "shifted" so that the (i, j) entry is filled with the Catalan number Ci+j−1 then the determinant is still 1, regardless of the value of n. For example, for n = 4 we have Catalan number_sentence_78

Taken together, these two conditions uniquely define the Catalan numbers. Catalan number_sentence_79

History Catalan number_section_10

The Catalan sequence was described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles. Catalan number_sentence_80

The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle. Catalan number_sentence_81

The counting trick for Dyck words was found by Désiré André in 1887. Catalan number_sentence_82

In 1988, it came to light that the Catalan number sequence had been used in China by the Mongolian mathematician Mingantu by 1730. Catalan number_sentence_83

That is when he started to write his book Ge Yuan Mi Lu Jie Fa [The 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. Catalan number_sentence_84

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. Catalan number_sentence_85

For instance, Ming used the Catalan sequence to express series expansions of sin(2α) and sin(4α) in terms of sin(α). Catalan number_sentence_86

Generalizations Catalan number_section_11

See also Catalan number_section_12

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Catalan number.