  We obtain estimates for the Kolmogorov distance to appropriately chosen
gaussians, of linear functions \[ \sum_{i\in [n]^d} \theta_i X_i \] of random
tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and
exchangeable, and whose entries have bounded third moment and vanish on
diagonal indices. These estimates are expressed in terms of intrinsic (and
easily computable) parameters associated with the random tensor
$\boldsymbol{X}$ and the given coefficients $\langle \theta_i:i\in
[n]^d\rangle$, and they are optimal in various regimes.
  The key ingredient -- which is of independent interest -- is a combinatorial
CLT for high-dimensional tensors which provides quantitative non-asymptotic
normality under suitable conditions, of statistics of the form \[
\sum_{(i_1,\dots,i_d)\in [n]^d}
\boldsymbol{\zeta}\big(i_1,\dots,i_d,\pi(i_1),\dots,\pi(i_d)\big) \] where
$\boldsymbol{\zeta}\colon [n]^d\times [n]^d\to\mathbb{R}$ is a deterministic
real tensor, and $\pi$ is a random permutation uniformly distributed on the
symmetric group $\mathbb{S}_n$. Our results extend, in any dimension $d$,
classical work of Bolthausen who covered the one-dimensional case, and more
recent work of Barbour/Chen who treated the two-dimensional case.
