Number of Cattleya

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Catalan number CattleyaTurn from:Http://www.mathoe.com/dispbbs.asp?boardid=89&replyid=46004&id=34522&page=1&skin=0&Star =2

About the Cattleya number of extensions:
1. (n-m+1)/(n+1) *c (n+m,n)
2.C[N+M][N]-C[N+M][M-1]
Catalan,eugene,charles, a Belgian mathematician of Cattleya (1814~1894), was born in Bruges (Brugge) and studied at the Paris Integrated Engineering school in the early years. He was a professor of mathematics at the University of Liege (Liege) in 1856 and was selected as a Fellow of the Brussels Academy of Sciences.

Cattleya has published more than 200 kinds of mathematics in various fields throughout his life. In the differential geometry, he proves the so-called Cattleya theorem: When a straight curve is a plane and a normal spiral surface, he can only be a real minimal surface. He also and Jacobian (Jacobi,c G. J) at the same time, the variable substitution problem of multiple integrals is solved, and the relevant formulas are established.

In 1842, he proposed a conjecture that the equation xz-yt=1 has no positive integer solution greater than 1 unless the trivial case 32-23= 1. The problem has not been solved yet.

(Mathoe Note: That is, except 8, 9, these two consecutive positive integers are positive integer operational outside, there is no other.) 1962 Chinese mathematician Koering the extremely consummate method to prove that there is no three consecutive positive integers, they are the power of positive integers, and the equation x2-yn=1,n>1,xy≠0 no positive integer solution. It is also proved that if the Cattleya conjecture is not established, its smallest counter-example ismore than ten. )

In addition, Cattleya also made some contributions in function theory, Bernoulli number and other fields.

Cattleya the series Cnby solving the convex n-side-shaped split.

The convex n+2 edges are divided into non-overlapping triangles with their n-1 diagonal lines, and the total number of such divisions is Cn.

To commemorate Cattleya, people use "Cattleya number" to name this series.

It is said that the final expression of dozens of seemingly unrelated combinatorial counting problems is in the form of Cattleya numbers.

Cattleya number in mathematics competitions, information science competitions, combinatorial mathematics, computer programming, etc. will have its different aspects of the introduction.

Number of first few Cattleya: Specify c0= 1, and

C1=1,c2=2,c3=5,c4=14,c5= 42,

C6=132,c7=429,c8=1430,c9=4862,c= 16796,

COne =58786,c =208012,c =742900,c =2674440,c 9694845.

Recursive formulas


There is a total of 2n dots labeled 1,2,3,4,......,2n on the circumference, and this 2n point pair can be connected to n strings, and these chords 22 do not intersect in the number of Cattleya cn.

2003 Zhejiang Provincial Primary School Mathematics summer Camp Contest Test the problem: the circumference of 10 points can be connected to neither intersect, there is no public end of the 5 line segments, different joint law altogether _____ species.

A: The number of methods is Cattleya c5= 42, this question is included in the Tan editor of the Knowledge Publishing house published the "number of Olympic Games intensive training" primary six grade book "Counting Problem" topic.


A total of six types, the 1th category there are 5 kinds of Lian, the 2nd class has 2 kinds of continuous law, 3rd class has 10 kinds of continuous law, 4th class has 10 kinds of continuous law, 5th class has 10 kinds of connection law, 6th class has 5 kinds of connection method. There are 42 kinds of connection laws.


1994 "Primary Mathematics" prize Knickknack contest: Amusement Park ticket 1 Yuan One, each one limited purchase. Now there are 10 children lined up to buy tickets, of which 5 children each have only 1 yuan a note, the other 5 children each have only 2 Yuan bill one, the conductor did not prepare change. Q: How many queuing methods are there to make the ticket clerk always find the change?

(This problem is also a number of Olympic data included as an example or exercise, "Hua School mathematics textbook" Elementary School six-year Book of Thinking training also received this question)

A: Now take 1 Yuan 5 children as the same, take 2 Yuan 5 children also as the same, using our common "dot accumulation method":

Each small section of the figure shows a child with 1 dollars, each small vertical section of the child to take 2 yuan, asked from A to B in the process of any point in the grid is not less than the number of vertical segments: Take 1 yuan to first, and the number of people can not be less than 2 yuan, that is, can not cross the diagonal ab: Each point is the number of points from a to this point. The number of methods to take the method from A to B. Per-point accumulation can be calculated as 42, i.e. Cattleya number c5= 42.



And because each child is not the same, so a total of 42x5! x5! =42x120x120=604800 of the situation.

If the problem of 10 people, take 1 yuan 5 people, take 2 yuan 5 people to a total of 2n individuals, 1 yuan of n people, 2 Yuan of n people, then the number of queuing methods to meet the requirements are:






Another example of a Cattleya number:

A two people race table tennis, the final result is 20:20, ask the game process a always lead B scoring situation of the number of species.

That is, a in the process of getting 1 points to 19 points is always ahead of B, the number of species is Cattleya number






Another example of a Cattleya number

After dinner, the elder sister washes the dishes, the sister washes the elder sister's bowl one by one into the cupboard stacks into a pile. There are a total of n different bowls, before washing is stacked into a pile, perhaps because the little sister to play and make the bowl into the cupboard, the elder sister will wash the bowls in the side, asked: how many kinds of bowls of the little sister can be the way?

A: It is the nth Cattleya number cn.

Another example of a Cattleya number

A car team on the narrow road, not overtaking, but can enter a dead end to refuel, and then jump in the queue, a total of n cars, asked how many different ways to make the team out of the city?

A: It is the nth Cattleya number cn.

Number of Cattleya

Verification: Cattleya number cn is an integer.

Prove:

① integer function Inequality: For any real number x, Y has [x+y]≥[x]+[y]. Here [x] represents the largest integer not greater than the real x.

Solution: By definition x≥[x] ... (1)

Y≥[y] ... (2) The above two-type addition, get: X+y≥[x]+[y],

The upper type is rounded up again: [x+y]≥[[x]+[y]]=[x]+[y], i.e. [x+y]≥[x]+[y].

②1000! The number of the end 0 249 . (There are now 100 of primary School Olympiad Books!) There are a few zeros at the end of the topic: 24)

Solution: 1000÷5=200,

200÷5=40,

40÷5=8,

8÷5=1 ... 3

The above quotient add, that is 1000! The number of the end 0 is =200+40+8+1=249.

③n! The number of powers of the mass factor p in the factorization type:


............ (1)

K! The power number of the mass factor p in the decomposition of qualitative factors

............ (2)

(N-K)! The power number of the mass factor p in the decomposition of qualitative factors


............ (3)

In this case, the Sigma summation is used in infinite form, but each item after a certain item is 0, and for unification, it is written in "∞" form.

④ combination number is an integer

Solution:

⑤ Cattleya number is an integer

⑥ Cattleya number is another proof of an integer

④ combination number is an integer




⑤ Cattleya number is an integer



⑥ Cattleya number is another proof of an integer





14 methods of convex hexagon split into triangles are Cattleya number C4







From the lower left (0,0) to the upper right corner (+), only allow up, to the right, but not allowed to cross the diagonal number of methods is 14, is the number of Cattleya C4



1936 the 40th Hungarian Olympiad Mathematics Competition 1th question has Catalan the identity type proof.





1979 21st International Mathematics Olympiad The 1th question of the application of a Cattleya identity






This question was adapted from the 1th Hungarian-Israeli mathematical contest in 1989.

Transferred from:http://www.mathoe.com/dispbbs.asp?boardid=89&replyid=46004&id=34522&page=1&skin=0& star=2

Number of Cattleya (RPM)

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