Introduction to "catlan number"

Introduced: catlan number.

Catalan number, also known as catalan number, is a series that often appears in various counting problems in composite mathematics.

-- Baidu encyclopedia

I. calculation formula. Let H (0) = 1, H (1) = 1, and the catalan number satisfy the recursive formula: H (n) = H (0) * H (n-1) + H (1) * H (n-2) +... + H (n-1) * H (0) (N> = 2)Example: H (3) = H (0) * H (2) + H (1) * H (1) + H (2) * H (0) = 1*2 + 1*1 + 2*1 = 5 2: code implementation.

# Include <bits/stdc ++. h> int N, F [30]; int main () {scanf ("% d", & N); F [0] = 1, F [1] = 1; for (INT I = 2; I <= N; I ++) for (Int J = 0; j <I; j ++) f [I] + = f [J] * f [i-j-1]; // catlan number printf ("% d", F [N]); Return 0 ;}

Recursive algorithms are used here.

It can also be implemented using recursion:

# Include <bits/stdc ++. h> using namespace STD; int H (int n)
{If (n = 0 | n = 1) return 1; return H (n-1) * (4 * N-2)/(n + 1 ); // formula} int main () {int N; CIN> N; cout <H (n); Return 0 ;}

 

3. Application.

· Description

Luogu p1044 stack [original question link]

· Resolution

The description of this question is very simple. N numbers are added to the stack sequentially, And the stack can be randomly output. There are several possibilities.

DFS can be implemented, but when you think carefully, you will find a simpler method.

Let's create array F.F [I] indicates the full possibility of I entry and exit.

Then f [0] = 1, F [1] = 1; (there is no other possibility of 0 or 1 .)

If X is the last of the current output stack sequence, X has n values.

Since X is the last outbound stack, you can divide the number of outbound stacks into two parts.

• Smaller than X
• Larger than X

A number smaller than X has a X-1, so all of these numbers may be f [x-1]

There are n-X numbers greater than X, so the full output stack of these numbers may be f [n-x]

These two partsMutual Influence, SoAll possible values of X are f [x-1] * f [n-X].

In addition, X has n values, so:

Ans = f [0] * f [n-1] + F [1] * f [N-2] +... + F [n-1] * f [0];

Obviously --Catlands!

The data scope of the question is very small, so you can paste the template directly.

Iv. Summary. So, let's do it first. The Catalan number has the following usage:

• Brackets matrix concatenation:P = A1 × a2 × A3 × ...... ×An: According to the multiplication combination law, without changing its order, we only use parentheses to represent the product of pairs. How many methods are there to enclose? (H (n)
• The given nodes form a binary search tree:How many different binary search trees can a given n nodes constitute? (Can constitute H (N) (the subscript of this formula starts from H (0) = 1)

N correct matching numbers of parentheses:Given n pairs of parentheses, evaluate the number of strings correctly matched by parentheses. (H (n)

· Classic exercise

P1722 matrix II [original question link]

 

Introduction to "catlan number"