Quick question. I am struggling to understand the definition of arm and leg length of a cell in a standard Young Tabluea. I am not sure just how to fill in the boxes. As an example I am considering $ \lambda = (9,6,6)$. Or $9+6+6=21$. If you want to illustrate with a smaller example that is fine. I understand the arm length is the number of boxes to the "right" and leg length is the numbers of boxes "below".
2025-01-13 05:55:28.1736747728
Arm and leg length of standard Young Tableux
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Suppose our diagram is of a partition $\lambda=(\lambda_1,\ldots,\lambda_k)$. The arm length of the box in row $i$ and column $j$ is $\lambda_i-j$. The leg length of the same box is the arm length in the transpose. Without referring to the transpose a formula is $$\#\{p>i:\lambda_p\geq j\}$$