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Catalan's Triangle
In combinatorial mathematics, Catalan's triangle is a number triangle whose entries C(n,k) give the number of strings consisting of ''n'' X's and ''k'' Y's such that no initial segment of the string has more Y's than X's. It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan. Bailey shows that C(n,k) satisfy the following properties: # C(n,0)=1 \text n\geq 0 . # C(n,1)=n \text n\geq 1 . # C(n+1,k)=C(n+1,k-1)+C(n,k) \text 1 n it is impossible to form a path that does not cross the constraint, i.e. C_(n,k)= 0 . (3) when m\leq k\leq n+m-1 , then C_(n,k) is the number of 'red' paths \left(\begin n+k\\ k \end\right) minus the number of 'yellow' paths that cross the constraint, i.e. \left(\begin (n+m)+(k-m)\\ k-m \end\right) = \left(\begin n+k\\ k-m \end\right). Therefore the number of paths from (0,0) to (k, n) that do not cross the constraint n - k + m - 1 = 0 is as indicated in the formula in the previous section "''Generalization''". Proof 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
