  Littlewood-Richardson coefficients are nonnegative integers that play a
pivotal role in combinatorial algebra, bridging the representation theory of
symmetric and general linear groups, the Chow rings of Grassmannians, and
eigenvalues of Hermitian matrices. Littlewood-Richardson rules are positive
combinatorial formulas for Littlewood-Richardson coefficients that manifest
their nonnegativity. More generally, Chow rings of flag varieties have bases of
Schubert cycles $\sigma_u$, indexed by permutations. Here, the
Littlewood-Richardson coefficients describe special products $\sigma_u \cdot
\sigma_v$ where $u$ and $v$ are $p$-Grassmannian permutations.
  Building on work on Wyser, we introduce backstable clans to prove a "dual"
Littlewood-Richardson rule for the product $\sigma_u \cdot \sigma_v$ when $u$
and $v$ are inverse to $p$-Grassmannian permutations. Moreover, our positive
combinatorial rule extends to the case where $u$ is $p$-inverse Grassmannian
and $v$ is $q$-inverse Grassmannian. We further give a positive combinatorial
rule for $\sigma_u \cdot \sigma_v$ in the case that $u$ is $p$-inverse
Grassmannian and $v$ is covered in weak Bruhat order by a $q$-inverse
Grassmannian permutation.
