Pascal's Triangle for non-commutative?

In summary, The conversation discusses the existence of a non-commutative version of Pascal's triangle for operators, particularly in the context of brah-ket notation. It is suggested that the middle coefficients in Pascal's triangle may correspond to cross-terms split between their orderings, but it is noted that these terms may not always appear in the expansion. The potential difficulty of generalizing Pascal's triangle with non-commutative numbers is also mentioned.
  • #1
Pythagorean
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I was curious if there was a non-commutative version of Pascal's triangle for operators (such as those used in brah-ket notation)

The important note is that (a + b)^2 = a^2 + b^2 + ab +ba
where ba != ab
 
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  • #2
I'm guess if it's an even power such as (x + y)^4

you can always assume the cross-terms are split between their orderings, so from pascal's triangle, which gives the coefficients 1 4 6 4 1, the middle coefficient corresponds to 3x^2y^2 + 3y^2x^2.

Not quite sure how to handle odd degrees without doing pages of algebra.

I'm posting in this forum because tensors have the non-commutative property and I happen to be applying brah-ket notation so I thought it fit. Apologies if not.
 
  • #3
This is interesting.

Going by (a+b)^3 = a^3 + a^2*b+aba + ab^2 + ba^2 + bab + b^2*a + b^3

It would seem that the only possible row in the "triangle" for general noncommuntative numbers would be "1 1 1 1 1 1 1 1". The exact form may depend on just how uncommutative the number are.
 
  • #4
ObsessiveMathsFreak said:
This is interesting.

Going by (a+b)^3 = a^3 + a^2*b+aba + ab^2 + ba^2 + bab + b^2*a + b^3

It would seem that the only possible row in the "triangle" for general noncommuntative numbers would be "1 1 1 1 1 1 1 1". The exact form may depend on just how uncommutative the number are.

wow, that looks like it may be difficult to generalize with something like Pascal's triangle. Maybe not the coefficients themselves, but the degree of each term. You'd have to have two degree functions for each variable (i.e. one for <x| and one for |x>)
 
  • #5
The point of Pascal's triangle is that the i,j entry simply counts the number of ways you can order i "x"s and j-i "y"s. If your multiplication is not commutative, those do not add. All terms are distinct. That's why ObsessiveMathFreak says you just get "1 1 1 ...".
 
  • #6
Pythagorean said:
I'm guess if it's an even power such as (x + y)^4

you can always assume the cross-terms are split between their orderings, so from pascal's triangle, which gives the coefficients 1 4 6 4 1, the middle coefficient corresponds to 3x^2y^2 + 3y^2x^2.
Just to make sure it's said... those terms do not appear in the expansion of (x+y)^4. (except possibly for special cases)
 

Related to Pascal's Triangle for non-commutative?

1. What is Pascal's Triangle for non-commutative?

Pascal's Triangle for non-commutative is a mathematical concept that extends the traditional Pascal's Triangle to include non-commutative operations, meaning the order in which numbers are multiplied affects the outcome. It is a triangular array of numbers where each number is the sum of the two numbers above it, following a specific pattern.

2. How is Pascal's Triangle for non-commutative different from the traditional Pascal's Triangle?

The main difference between the two is that in traditional Pascal's Triangle, the sum of two numbers above a certain number is always the same regardless of the order in which they are multiplied. However, in Pascal's Triangle for non-commutative, the order of multiplication affects the outcome, resulting in a different triangle with unique properties.

3. What are some real-world applications of Pascal's Triangle for non-commutative?

Pascal's Triangle for non-commutative has applications in fields such as computer science, coding theory, and quantum mechanics. It is used to solve problems related to non-commutative operations, such as finding the number of unique paths in a network or determining the efficiency of different coding schemes.

4. How is Pascal's Triangle for non-commutative calculated?

To calculate Pascal's Triangle for non-commutative, the first row is always 1, and each subsequent row is calculated by multiplying the previous row with itself, starting from the left. For example, the third row would be (1, 1) x (1, 1) = (1, 2, 1). The triangle continues infinitely, with each row adding an additional number to the end.

5. What are some interesting properties of Pascal's Triangle for non-commutative?

One interesting property is that the sum of the numbers in each row equals 2 to the power of n, where n is the row number. Additionally, the numbers in the triangle can be used to represent coefficients in binomial expansions with non-commutative variables. This allows for the calculation of complex equations involving non-commutative operations.

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