Skew shape formula

Statement

There is a generalization of this formula for skew shapes,

$$s_{\lambda/\mu}(1,t,t^2,\cdots) = \sum_{S \in E(\lambda/\mu)} \prod_{(i,j) \in \lambda\setminus S} \frac{t^{\lambda_j'-i}}{1-t^{h(i,j)}}$$

where the sum is taken over excited diagrams of shape $\lambda$ and boxes distributed according to $\mu$.

A variation on the same theme is given by Okounkov and Olshanski of the form

$$\frac{\dim\lambda/\mu}{\dim\lambda}=\frac{s^*_\mu(\lambda)}{|\lambda|!/|\mu|!},$$

where $s^*_\mu$ is the so-called shifted Schur function $s^*_\mu(x_1,\dots,x_n)=\frac{\det[(x_i+n-1)!/(\mu_j+n-j)!]}{\det[(x_i+n-i)!/(n-j)!]}$.