Pascal_like_triangles

2. Mathematical theory of Pascal like triangles.

In the 17th century Blaise Pascal discovered the Pascal’s triangle.

In the 21th century the authors discovered Pascal-like triangles that are generalizations of Pascal’s triangle.

By Theorem 2 U[p,n,m,v]　has the property that is very similar to nCm . Therefore for fixed natural numbers p,v the list { F[p,n,m,v]= , m ≤ n and n = 1,2,3,...} can form a Pascal's triangle-like pattern of fractions when we arrange them to make a triangle. See the following Example 1.

Example 1. The purpose of this example is to observe a Pascal's triangle-like pattern of fractions.

To do this we have to make a triangle. Here we let p = 2 and v = 1, and make F[p,n,m,v] for any natural number n,m with n ≤ 7 and m ≤ n. Then we get a triangle(See Figure 1). If we calculate F[2,n,m,1] for each n and m, then we have the following triangle of fractions (See Figure 2).　

The pattern is quite obvious in Figure (2). For example look at 6th row.

F[2,6,2,1]= and F[2,6,3,1]= are the second and third ones in the row.

F[2,7,3,1]= = , which is the third one in the 7th row. This reminds us of

Pascal's triangle. By Theorem 1 and 2 there exists the same kind of relation among

F[p,n,m,v] ,F[p,n,m+1,v], F[p,n+1,m+1,v] for any natural number p,v,n and m with n ≥ m.

Figure(1)

Figure(2)

In this section we are going to study the mathematical background of Pascal like triangles.

We omit the proof of theorems and lemmas here. If you want to read proofs, please click “Proof.”

Definition 1: Let p,n,m be fixed natural number such that m ≤ n. We have players θ₁,θ₂,θ₃… , θp who are seated around a circle. The game begins with player θ₁. Proceeding in order, a box is passed from hand to hand. The box contains n numbers. The numbers in the box are assigned in numerical order, from 1 to n When a player receive the box, he draws a number from the box. The players cannot see the numbers when they draw numbers, and hence they draw at random. Once a certain number was taken from the box, that number will not be returned to the box. If any player gets a number x such that x ≤ m, he will be the loser of the game and the game ends.

We denote by R(n,m,y) the probability of the game ends in the y th round.

Lemma 1: For natural numbers n,m,y such that y≤n-m+1 we have

(1)

Proof.

We are going to use the floor function. For any real number x the function

gives the greatest integer less than or equal to x.

Theorem 1:

Proof.

Definition 2: We define

This number U(p,n,m,v) is important throughout this article. By Theorem 1 and Definition 3

Lemma 2:

　(3) U(p,n,n,1) = 1.

Proof.

In the following Theorem 2 we are going to prove that U(p,n,m,v) has the property that is very similar to the property of nCm . To prove this we have only to use the well known formula nCm-1 + rCm = r+1Cm for each term of U(p,n,m,v) .

Theorem 2: 　　　U(p,n,m,v) + U(p,n,m+1,v) = U(p,n+1,m+1,v).

Proof.