What is the Dimension of a Symmetric Tensor Vector Space?

[sgd37]

Homework Statement

Having a symmetric tensor $$S^{a_1 ...a_n}$$ forming a vector space $$V_n$$ with indices taking values from 1 to 3; what is the dimension of such a vector space?

Homework Equations

The Attempt at a Solution

essentially this reduces to picking a tensor of type $$S^{ \underbrace{1...1}_{s} \underbrace{2...2}_{r} \underbrace{3...3}_{t}}$$ with r+s+t =n and seeing how many non isomorphic combinations there are. I'm not that skilled at combinatorics unfortunately

 

[diazona]

So... if I'm understanding correctly, this is a slightly roundabout way to ask how many independent components the tensor $S$ has?

Think about it this way: you have a list of $n$ indices which you need to split into a set of 1's, a set of 2's and a set of 3's, in that order. How many places can you choose to put the split between the 1's and the 2's?

Then, given that choice, how many places can you choose to put the split between the 2's and the 3's?

 

[sgd37]

worked it out a different way, thanks all the same