,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 a ...
with ''n'' + 2 sides can be cut into
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
s by connecting vertices with non-crossing
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
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 ''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
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 is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s 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
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
that all of the ''free''
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have ...
s 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
A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains con ...
and ''n'' − 1 votes,
is the number of possible voting outcomes, given that at any node you can split your votes anyway you want.
*
is the number of length sequences that start with
, and can increase by either
or
, or decrease by any number (to at least
). For
these are
. 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
is missing.
Proof of the formula
There are several ways of explaining why the formula
:
solves the combinatorial problems listed above. The first proof below uses a
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
. 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 othe ...
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 ...
:
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
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
for the Catalan numbers is defined by
:
The recurrence relation given above can then be summarized in generating function form by the relation
:
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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
. On the one hand, the recurrence relation uniquely determines the Catalan numbers; on the other hand, interpreting as a
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
of ''c'' and using the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
, the generating function relation can be algebraically solved to yield two solution possibilities
:
or
.
From the two possibilities, the second must be chosen because only the choice of the second gives
:
.
The square root term can be expanded as a power series using the identity
:
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
:
and substituting this power series into the expression for ''c''(''x''), the expansion simplifies to
:
.
Let
, so that
:
and because
(see 'proof of recurrence' above)
we have
:
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
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,
:
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
monotonic paths, we obtain the desired formula
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
. Let
be the first that brings an initial subsequence to equality, and configure the sequence as
. The new sequence is
.
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
:
.
Write
Because
then
:
Applying the recursion with
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'' =
1 ">nbsp;''c''1 ''c''
2, summing over the possible spots to place the closing bracket immediately gives the recursive definition
:
Let ''b'' stand for a ''balanced'' string of length 2''n''—that is, containing an equal number of "
and ", so
. As before, any balanced string can be uniquely decomposed into either
nbsp;''c'' nbsp;''b'' or ] ''c''
' B_ = 2\sum_^n B_i C_.
Any incorrect balanced string starts with ''c'' ">nbsp;''b'', so
:
Any incorrect balanced string starts with ''c'' and the remaining string has one more
than so
:
Also, from the definitions, we have:
:
Therefore
:
:
:
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
X's and
Y's, where
, has precisely
dominating cyclic permutations. To see this, arrange the given sequence of
X's and Y's in a circle. Repeatedly removing XY pairs leaves exactly
X's. Each of these X's was the start of a dominating cyclic permutation before anything was removed.
For example, consider
. This is dominating, but none of its cyclic permutations
,
,
and
are.
In particular, when
, there is exactly one dominating cyclic permutation. Removing the leading X from it (a dominating sequence must begin with X) leaves a Dyck sequence. Since there are
in total, and each one belongs to an equivalence class of size ''2n+1'' (because ''n'', ''m'' and ''2n+1'' are pairwise coprime), we have
distinct cycles of
X's and
Y's, each of which corresponds to exactly one Dyck sequence, hence
counts Dyck sequences.
Hankel matrix
The ''n''×''n''
Hankel matrix whose (''i'', ''j'') entry is the Catalan number ''C''
''i''+''j''−2 has
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1, regardless of the value of ''n''. For example, for ''n'' = 4 we have
:
Moreover, if the indexing is "shifted" so that the (''i'', ''j'') entry is filled with the Catalan number ''C''
''i''+''j''−1 then the determinant is still 1, regardless of the value of ''n''.
For example, for ''n'' = 4 we have
:
Taken together, these two conditions uniquely define the Catalan numbers.
Another feature unique to the Catalan–Hankel matrix is the determinant of the ''n''×''n'' submatrix starting at ''2'' has determinant ''n'' + 1.
:
:
:
:
et cetera.
History
The Catalan sequence was described in the 18th century by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
, 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. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
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
and
in terms of
.
Generalizations
The Catalan numbers can be interpreted as a special case of the
Bertrand's ballot theorem. Specifically,
is the number of ways for a candidate A with ''n+1'' votes to lead candidate B with ''n'' votes.
The two-parameter sequence of non-negative integers
is a generalization of the Catalan numbers. These are named super-Catalan numbers, per
Ira Gessel
Ira Martin Gessel (born 9 April 1951 in Philadelphia, Pennsylvania) is an American mathematician, known for his work in combinatorics. He is a long-time faculty member at Brandeis University and resides in Arlington, Massachusetts.
Education ...
. These should not confused with the
Schröder–Hipparchus numbers, which sometimes are also called super-Catalan numbers.
For
, this is just two times the ordinary Catalan numbers, and for
, the numbers have an easy combinatorial description.
However, other combinatorial descriptions are only known
for
and
,
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, 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 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 representatio ...
, it is the number of anti-chains (or order ideals) in the poset of positive roots. The classical Catalan number
corresponds to the root system of type
. 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 ''k''-fold convolution is:
:
The Catalan numbers are a solution of a version of the
Hausdorff moment problem.
See also
*
Associahedron
*
Bertrand's ballot theorem
*
Binomial transform
*
Catalan's triangle
*
Catalan–Mersenne number
*
Fuss–Catalan number In combinatorial mathematics and statistics, the Fuss–Catalan numbers are numbers of the form
:A_m(p,r)\equiv\frac\binom = \frac\prod_^(mp+r-i) = r\frac.
They are named after N. I. Fuss and Eugène Charles Catalan.
In some publicati ...
*
List of factorial and binomial topics
*
Lobb numbers
*
Narayana number
*
Schröder–Hipparchus number
*
Tamari lattice
*
Wedderburn–Etherington number The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington.. and Joseph Wedderburn. that can be used to count certain kinds of binary trees. The first few numbers in the sequence are
:0, 1, 1, 1, 2, 3, ...
Notes
References
* Stanley, Richard P. (2015), ''Catalan numbers''. Cambridge University Press, .
*
Conway 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 an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
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
Articles containing proofs