Catalan's Problem

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 (1814–1894).

The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by

:$C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0.$

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

:1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... .

Properties

An alternative expression for ''C''_''n'' is
:$C_n = -$ for $n\ge 0,$
which is equivalent to the expression given above because $\tbinom=\tfrac\tbinomn$. This expression shows that ''C''_''n'' 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.

The Catalan numbers satisfy the recurrence relations
:$C_0 = 1 \quad \text \quad C_=\sum_^C_iC_\quad\textn\ge 0$
and
:$C_0 = 1 \quad \text \quad C_ = \fracC_n.$

Asymptotically, the Catalan numbers grow as
$C_n \sim \frac,$
in the sense that the quotient of the ''n''th Catalan number and the expression on the right tends towards 1 as ''n'' approaches infinity. This can be proved by using the asymptotic growth of the central binomial coefficients, by Stirling's approximation for $n!$, or via generating functions.

The only Catalan numbers ''C''_''n'' that are odd are those for which ''n'' = 2^''k'' − 1; all others are even. The only prime Catalan numbers are ''C''₂ = 2 and ''C''₃ = 5.

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 $\sum_^\infty \frac = 2$.

The latter representation is closely connected to Wigner's semicircle law for eigenvalue distribution of random symmetric matrices.

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 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 ''C''₃ = 5 and ''C''₄ = 14.

* ''C''_''n'' is the number of Dyck words of length 2''n''. 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.

* Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, ''C''_''n'' counts the number of expressions containing ''n'' pairs of parentheses which are correctly matched:

((()))     ()(())     ()()()     (())()     (()())

* ''C''_''n'' 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, as in the matrix chain multiplication problem). For ''n'' = 3, 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, with each correctly matched bracketing describing an internal node. It follows that ''C''_''n'' is the number of full binary trees with ''n'' + 1 leaves, or, equivalently, with a total of ''n'' internal nodes:

Also, the interior of the correctly matching closing Y for the first X of a Dyck word contains the description of the left subtree, with the exterior describing the right subtree.
* ''C''_''n'' is the number of non-isomorphic ordered (or plane) trees with vertices. See encoding general trees as binary trees.
* ''C''_''n'' 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".

The following diagrams show the case ''n'' = 4:

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 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 ''C''_''n''. The following hexagons illustrate the case ''n'' = 4:

* ''C''_''n'' is the number of stack-sortable permutations of . 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.
* ''C''_''n'' is the number of permutations of 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.
* ''C''_''n'' is the number of noncrossing partitions of the set . ''A fortiori'', ''C''_''n'' never exceeds the ''n''th Bell number. ''C''_''n'' is also the number of noncrossing partitions of the set 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.
* ''C''_''n'' is the number of ways to tile a stairstep shape of height ''n'' with ''n'' rectangles. Cutting across the anti-diagonal and looking at only the edges gives full binary trees. The following figure illustrates the case ''n'' = 4:

* ''C''_''n'' 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.

* ''C''_''n'' 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, ..., 2''n'' 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.
* ''C''_''n'' is the number of semiorders on ''n'' unlabeled items.

* Given an infinite perfect binary decision tree and ''n'' − 1 votes, $C_n$ is the number of possible voting outcomes, given that at any node you can split your votes anyway you want.
* $C_n$ 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 ''0''. An X increases the counter by ''1'' and a Y decreases it by ''1''. 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. 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.

First proof

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

:$C_0 = 1 \quad \text \quad C_=\sum_^n C_i\,C_\quad\textn\ge 0.$

For example, every Dyck word ''w'' of length ≥ 2 can be written in a unique way in the form
:''w'' = X''w''₁Y''w''₂
with (possibly empty) Dyck words ''w''₁ and ''w''₂.

The generating function 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. On the one hand, the recurrence relation uniquely determines the Catalan numbers; on the other hand, interpreting as a quadratic equation of ''c'' and using the quadratic formula, 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 choice of the second gives

:$C_0 = \lim_ c(x) = 1$.

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

:$\sqrt = \sum_^\infty y^n = \sum_^\infty \frac y^n$

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.

Setting ''y'' = −4''x'' gives

:$\sqrt = \sum_^\infty \frac x^n = 1 - \sum_^\infty \frac x^n$

and substituting this power series into the expression for ''c''(''x''), the expansion simplifies to

:$\sum_^\infty \frac x^$.

Let $N=n-1$, so that

:$\sum_^\infty \frac x^N$

and because $\frac = C_N$ (see 'proof of recurrence' above)

we have

:$c(x) = \sum_^\infty \frac.$

Second proof

We count the number of paths which start and end on the diagonal of a ''n'' × ''n'' grid. All such paths have ''n'' right and ''n'' up steps. Since we can choose which of the 2''n'' 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 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 2''n'' steps, there are now ''n'' + 1 up steps and ''n'' − 1 right steps. So, instead of reaching (''n'',''n''), all bad paths after reflection end at (''n'' − 1, ''n'' + 1). Because every monotonic path in the (''n'' − 1) × (''n'' + 1) 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 ''n'' X's and ''n'' 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 ''n'' + 1 appearing in the denominator of the formula for ''C''_''n''. 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 ''1'' 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 ''X'' the first such edge that is reached.
* Swap the portion of the path occurring before ''X'' with the portion occurring after ''X''.
In Figure 3, the black dot indicates the point where the path first crosses the diagonal. The black edge is ''X'', 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 ''3'' to ''2''. In fact, the algorithm causes the exceedance to decrease by ''1'' 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 passes from above the diagonal to below it - all the other vertical edges stay on the same side of the diagonal.

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. 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. 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 ''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. 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 $\textstyle$ monotonic paths, we obtain the desired formula $\textstyle C_n = \frac.$

Figure 4 illustrates the situation for ''n'' = 3. 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, ''C''₃ = 5, 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 ''C_n'' and ''C''_''n''+1.

Given a polygon ''P'' with ''n'' + 2 sides and a triangulation, mark one of its sides as the base, and also orient one of its 2''n'' + 1 total edges. There are (4''n'' + 2)''C''_''n'' such marked triangulations for a given base.

Given a polygon ''Q'' with ''n'' + 3 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 (''n'' + 2)''C''_''n'' + 1 such marked triangulations for a given base.

There is a simple bijection between these two marked triangulations: We can either collapse the triangle in ''Q'' whose side is marked (in two ways, and subtract the two that cannot collapse the base), or, in reverse, expand the oriented edge in ''P'' to a triangle and mark its new side.

Thus
:$(4n+2)C_n = (n+2)C_$.

Write $\textstyle\fracC_ = C_n.$

Because $(2n)!=(2n)!!(2n-1)!!=2^nn!(2n-1)!!$

then

:$\frac=2^n(2n-1)!!=(4n-2)!!!!.$

Applying the recursion with $C_1=1$ 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 ''n'' pairs of brackets. 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'' =  nbsp;''c''₁ ''c''₂, summing over the possible spots to place the closing bracket immediately gives the recursive definition
:$C_0 = 1 \quad \text \quad C_ = \sum_^n C_i\,C_\quad\textn\ge 0.$

Let ''b'' stand for a ''balanced'' string of length 2''n''—that is, containing an equal number of " and ", so $\textstyle B_n =$. As before, any balanced string can be uniquely decomposed into either nbsp;''c'' nbsp;''b'' or ] ''c''^'  nbsp;''b'', so

:$B_ = 2\sum_^n B_i C_.$

Any incorrect 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

:$2B_i-C_i$

\fracB_i,

:$C_i=B_i\left(2-\frac\right),$

:$C_i=\frac\binom.$

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