フック長の公式
For example, UUDDUD correspond to the tableaux with rows 125 and 346.
The hook formula gives another way of getting a closed formula for the Catalan numbers
C n = ( 2 n ) ! ( n + 1 ) ( n ) ⋯ ( 3 ) ( 2 ) ( n ) ( n − 1 ) ⋯ ( 2 ) ( 1 ) = ( 2 n ) ! ( n + 1 ) ! n ! = 1 n + 1 ( 2 n n ) {\displaystyle C_{n}={\frac {(2n)!}{(n+1)(n)\cdots (3)(2)(n)(n-1)\cdots (2)(1)}}={\frac {(2n)!}{(n+1)!n!}}={\frac {1}{n+1}}{\binom {2n}{n}}}