Let $Y=(y_{ij})_{n\times n}$ be a symmetric matrix with diagonal elements being $0$. Define the following quantity \begin{equation} V(y) = \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \sum_{\ell=1}^n y_{ij} y_{ik} y_{i\ell} y_{jk} y_{j\ell} y_{k\ell}. \end{equation} How to write $V(y)$ in matrix form?
(ps: Using trace, sum of rows/columns of matrix, element-wise product of matrix $\textit{etc.}$ are allowed.)
$ \def\dss{\mathop{\odot}\limits} \def\bbR#1{{\mathbb R}^{#1}} \def\a{\;{\rm and}\;}\def\b{\beta}\def\g{\gamma} \def\d{\delta}\def\t{\theta} \def\l{\lambda}\def\s{\sigma}\def\e{\varepsilon} \def\n{\nabla}\def\o{{\tt1}}\def\p{\partial} \def\D{{\cal D}}\def\F{{\cal F}}\def\G{{\cal G}} \def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)} \def\vec#1{\operatorname{vec}\LR{#1}} \def\diag#1{\operatorname{diag}\LR{#1}} \def\Diag#1{\operatorname{Diag}\LR{#1}} \def\trace#1{\operatorname{Tr}\LR{#1}} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\hess#1#2#3{\frac{\p^2 #1}{\p #2\,\p #3}} \def\c#1{\color{red}{#1}} \def\m#1{\left[\begin{array}{r}#1\end{array}\right]} \def\rc#1{\color{red}{#1}} \def\bc#1{\color{blue}{#1}} \def\gc#1{\color{green}{#1}} $Define the $12^{th}$ order analog of the Kronecker delta symbol $$\eqalign{ \D_{i\rc{j}w\bc{k}x\gc{\ell}\rc{m}\bc{p}\rc{n}\gc{r}\bc{q}\gc{s}} = \begin{cases} \o \quad{\rm if}\;(i=w=x) \\ \qquad\a(\rc{j=m=n}) \\ \qquad\a(\bc{k=p=q}) \\ \qquad\a(\gc{\ell=r=s}) \\ 0 \quad{\rm otherwise} \\ \end{cases} \\ }$$ and the sixth dyadic power of $Y$ as $$\eqalign{ Y^{\star 6} &= Y\star Y\star Y\star Y\star Y\star Y \\ }$$ where $(\star)$ denotes the dyadic/tensor product.
Then the sum in question could be written as $$\eqalign{ V &= \D \,:\,:\,:\,:\,:\,:\, Y^{\star 6} \\ &= \D \;\dss^{12}\; Y^{\star 6}\\ }$$ where the $12$ dots between the tensors denote contraction products.