# Bijection between Littlewood-Richardson tableaux and ordinary skew tableaux?

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?

## 1 Answer

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.