The Dimension of the Symmetric $k$-tensors I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the dimension of covariant $k$-tensors is $n^k$ but how can I eliminate non-symmetric the cases? I found this but I could not get the intution. Also, this blog post answers my question but I don't see why we put | between different indices. Any concrete example would also help such as the symmetric covariant 2-tensors in $\mathbb{R^3}$, as I asked in this thread. 

A basis for symmetric tensors, say $\text{Sym}_r(V)$ with $\{v_1,...,v_n\}$ a basis for $V$, is given by the symmetrizations of $\{v_{i_1}\otimes ... \otimes v_{i_r} \ | \ 1\leq i_1\leq...\leq i_r\leq n\}$.  You must count the number of non-decreasing sequences (repetitions allowed) of length $r$ with entries in $[1,n]$.  I've always heard the method of counting these referred to as stars and bars, i.e. counting the number of multisets of size $r$ with entries from $[1,n]$, and the answer you get is ${n+r-1\choose r}$.

You line up $r$ stars and insert $n-1$ bars, the first bar separating indicies 1 and 2, the second bar separating indicies 2 and 3, ..., the $(n-1)$st bar separating indicies $n-1$ and $n$.
For example, say $r=5$ and $n=3$.  Here are some example of non-decreasing sequences of length $r=5$ with entries from $\{1,2,3\}$:
\begin{align*}
11223 \ &: \ **|**|*\\
22333 \ &: \ |**|***\\
11122 \ &: \ ***|**|\\
22222 \ &: \ |*****|\\
\end{align*}
So there are $r+n-1$ ``things'' (stars and bars) and you're choosing $r$ of them to be stars (or $n-1$ of them to be bars).

As for why this determines a basis for symmetric tensors:  any pure tensor on the chosen basis determines a symmetric tensor via
$$
S(v_{i_1}\otimes ... \otimes v_{i_r})=\sum_{\pi\in S_n}v_{\pi(i_1)}\otimes ... \otimes v_{\pi(i_r)}
$$
and two pure tensors have the same symmetrization if their indices determine the same multiset (i.e. non-decreasing sequence as described above).  I'll leave it to the reader to show that these are independent and that they span the space of symmetric tensors.  (On a technical note, the symmetrization needs to be modified in non-zero characteristic and some sources might divide by $n!$.)
