フック長の公式
The figure on the right shows hook-lengths for all cells in the Young diagram λ of the partition
9 = 4 + 3 + 1 + 1. Then the number of standard Young tableaux d λ {\displaystyle d_{\lambda }} for this Young diagram can be computed as
d λ = 9 ! 7 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 1 ⋅ 1 ⋅ 1 = 216. {\displaystyle d_{\lambda }={\frac {9!}{7\cdot 5\cdot 4\cdot 3\cdot 2\cdot 2\cdot 1\cdot 1\cdot 1}}=216.}
The hook-length formula can be understood intuitively using the following heuristic, but incorrect, argument suggested by D. E. Knuth. Given that each element of a tableau is the smallest in its hook and filling the tableau shape at random, the probability that cell ( i , j ) {\displaystyle (i,j)} will contain the minimum element of the corresponding hook is the reciprocal of the hook length.