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.
Just to make sure it's said... those terms do not appear in the expansion of (x+y)^4. (except possibly for special cases)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.
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.
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.
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.
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.
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.