Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ (where $\nu$,$\mu$ and $\lambda$ are integer partitions such that $|\nu| + |\mu| = |\lambda|$) are well-known coefficients appearing in various contexts.

In terms of Schur functions, which form a basis of the symmetric functions, the coefficients are the multiplicative constants, meaning that

$$ s_{\nu}s_{\mu} = \displaystyle \sum_{\lambda} c_{\nu\mu}^{\lambda} s_{\lambda}. $$

This is the context in which the rule was first stated. As Schur functions are the characters of irreducible representations of $\text{GL}_n$, it means that if we denote $V_{\lambda}$ for the irreducible representation of $\text{GL}_n$ associated to $\lambda$, then:

$$ V_{\nu} \otimes V_{\mu} \cong \displaystyle \bigoplus_{\lambda} (V_{\lambda})^{\oplus c_{\nu\mu}^{\lambda}} $$

The transition between those two contexts is relatively easy.

There is also another context closely related to the two above, which is the representations the symmetric group. Via the Schur-Weyl duality, the above results implies that if $S^{\lambda}$ is the Specht module associated to $\lambda$ (which are the irreducible representations of $\mathfrak{S}_{|\lambda|}$), then

$$ \left( S^{\nu} \otimes S^{\mu} \right) \big\uparrow_{\mathfrak{S}_{|\nu|} \times \mathfrak{S}_{|\mu|}}^{\mathfrak{S}_{|\nu|+|\mu|}} \cong \bigoplus_{\lambda} \left( S^{\lambda} \right)^{\oplus c_{\nu\mu}^{\lambda}} $$

Using Schur's lemma, the latter means that

$$ \text{dim} \left( \text{Hom}_{\mathfrak{S}_{|\nu|+|\mu|}} \left( S^{\lambda}, \left( S^{\nu} \otimes S^{\mu} \right) \big\uparrow_{\mathfrak{S}_{|\nu|} \times \mathfrak{S}_{|\mu|}}^{\mathfrak{S}_{|\nu|+|\mu|}} \right) \right) = c_{\nu\mu}^{\lambda} $$

My question is: is there any construction of the Littlewood-Richardson coefficients which is stated only in terms of the symmetric group and that gives the last equality? For example, Specht modules is the span of certain elements of the symmetric group algebra, called polytabloids. Can we calculate the Littlewood-Richardson coefficients using those?