TITLE: Bijection between Littlewood-Richardson tableaux and ordinary skew tableaux?
QUESTION [5 upvotes]: This question related to this,
but since that question does not ask the same thing, the answer to that question do not fully answer this.
Question: Is there a bijection between the set of Littlewood-Richardson tableaux of 
skew shape $\tau/\sigma$ and weight $\rho$ and 
the set of ordinary skew tableaux of shape $\lambda/\mu$ and weight $\nu$
for some $(\lambda,\mu,\nu)$?
In the question above, Richard Stanley gives a nice bijection 
in the case when $\tau/\sigma$ and $\rho$ have a quite special form,
(the LR-tableaux are essentially of the form of a disjoint union of
an ordinary tableau, and horizontal strips), and these can be set in bijection with
a set of non-skew tableaux with fixed shape and weight.
So, if we also allow to map to skew tableaux, can we 
extend this to cover all LR-tableaux, in some way?
For example, if we have that LR-tableaux $(\tau/\sigma,\rho)$
can bijectively be mapped to semi-standard (non-skew) tableaux with weight $\tau-\sigma$ and shape $\rho$, but this only works under the condition that $\tau_{i+1}=\sigma_{i}$ for all $i\geq 0.$ By allowing to map to skew tableaux as well, can we overcome this restriction?

REPLY [2 votes]: There has not been any answer to this, so I'll give a partial answer for the moment:
There is a bijection described by Littlewood and Richardson in the following sense:
$$K_{\lambda/\mu,\pi} \leftrightarrow \bigsqcup_\nu c_{\lambda/\mu,\nu} \times K_{\nu,\pi}$$
Here, I abuse notation and let $K$ and $c$ denote the sets of ordinary skew tableaux and Littlewood-Richardson tableaux.
This is the closest to a bijection that I can get as for now.