フック長の公式
In general, if A {\displaystyle A} is any poset with n {\displaystyle n} elements, the generating function for reverse A {\displaystyle A} -partitions is
P ( x ) ( 1 − x ) ( 1 − x 2 ) ⋯ ( 1 − x n ) {\displaystyle {\frac {P(x)}{(1-x)(1-x^{2})\cdots (1-x^{n})}}}
where P ( x ) {\displaystyle P(x)} is a polynomial such that P ( 1 ) {\displaystyle P(1)} is the number of natural labelings of A {\displaystyle A} .
In the case of a partition λ {\displaystyle \lambda } , we are considering the poset in its cells given by the relation
( i , j ) ≤ ( i ′ , j ′ ) ⟺ i ≤ i ′ and j ≤ j ′ {\displaystyle (i,j)\leq (i',j')\iff i\leq i'\qquad {\textrm {and}}\qquad j\leq j'} .
So a natural labeling is simply a standard Young tableau, i.e. P ( 1 ) = f λ {\displaystyle P(1)=f^{\lambda }}