Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2007 / B / 6 / a / Solution

For an integer partition $\lambda =({\lambda }_1,\dots ,{\lambda }_r)$, its Young diagram has ${\lambda }_i$ left-justified cells in row $i$. A Young tableau of shape $\lambda$ is a filling of its cells. Under the usual standard numbering convention, a standard Young tableau uses each integer $1,\dots ,n$ once and increases strictly along every row and down every column. For a different fixed ordered set of distinct labels, the same inequalities define the corresponding ordered-label version, which becomes standard after replacing its labels by their ranks.