  We give a new interpretation of the shifted Littlewood-Richardson
coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions).
The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur
$Q$-function $Q_\lambda$ into the sum of Schur functions $Q_\lambda =
2^{l(\lambda)}\sum_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special
case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length
$l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the
cardinal of a subset of a set that counts Littlewood-Richardson coefficients
$c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish
connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$.
More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and
$g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that
$g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some
conjectures on our combinatorial models which would imply this inequality if it
is valid.
