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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?

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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.

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