On the Shifted Littlewood-Richardson Coefficients and Littlewood-Richardson Coefficients

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 $g_{\lambda \mu }\le c_{{\mu }^t\mu }^{\tilde{\lambda }}$. We conjecture that $g_{\lambda \mu }^2\le c_{{\mu }^t\mu }^{\tilde{\lambda }}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.