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