= Hook-length formula
{title2=$f^\lambda=n!/\prod_{(i,j)\in Y(\lambda)}h_{i,j}(\lambda)$}
{wiki}
For any <partition of an integer> $\lambda\vdash n$, the number $f^\lambda$ of <standard Young tableaux> of shape $\lambda$ is
$$
f^\lambda=|\operatorname{SYT}(\lambda)|
=\frac{n!}{\displaystyle\prod_{(i,j)\in Y(\lambda)}h_{i,j}(\lambda)}.
$$
The numerator is a <factorial>, and the denominator is the <hook product of a partition>, using every cell's <hook length> once. The empty shape has one empty <standard Young tableau>, and the formula still holds with $0!=1$ and an empty product equal to one. By the <standard polytabloid basis>, $f^\lambda$ is also the <dimension> of the <Specht module> $S^\lambda$ over the <complex numbers>, or equivalently the degree $\chi^\lambda(1)$ of its <irreducible character>. For a teaching derivation of this <dimension> formula, see https://math.mit.edu/~etingof/representationtheorybook.pdf[Etingof et al., Introduction to Representation Theory, §5.17, Theorem 5.17.1].
In the <Young diagram> of $(3,2)$, the <hook length> array $[[4,3,1],[2,1]]$ gives
$$
H_{(3,2)}=4\cdot3\cdot1\cdot2\cdot1=24,\qquad
f^{(3,2)}=\frac{5!}{24}=\frac{120}{24}=5
=\dim_{\mathbb C}S^{(3,2)}.
$$
This agrees with the five explicit fillings in <standard Young tableau>. Its <conjugate partition> $(2,2,1)$ has the same <hook lengths> after transposition, hence also gives <dimension> five. At the extremes, for $n\geq1$, a single row $(n)$ and a single column $(1^n)$ each have <hook product> $n!$ and exactly one <standard Young tableau>; their <Specht modules> over the <complex numbers> are the <trivial representation> and <sign representation>, respectively.
Why the formula is beautiful: global compatibility of all the row and column inequalities reduces to a product of local <hook lengths>. It is useful when enumeration is already expensive: computing one <hook length> per cell replaces testing up to $n!$ numbered <Young tableaux>. For $(3,2)$, the proportion of the $120$ bijective fillings that are <standard Young tableaux> is $5/120=1/24$, the reciprocal of the <hook product>. This product does not justify treating overlapping hook conditions as independent; Grinberg explains that tempting but invalid argument in https://www.cip.ifi.lmu.de/~grinberg/t/24s/hooktalk.pdf[§1.3 of his teaching slides].
Historically, the modern hook-product statement was published by J. S. Frame, G. de B. Robinson and R. M. Thrall in 1954: https://doi.org/10.4153/CJM-1954-030-1[The Hook Graphs of the Symmetric Group, Canadian Journal of Mathematics 6, 316–324, Theorem 1]{ref}. Their manuscript was received on 1 September 1953, and their proof reorganizes an earlier <irreducible character> degree formula into the product of the <hook lengths>. The theorem unifies the counting and <representation> viewpoints developed earlier by Young. For proof routes within this wiki, see <trace computation of Specht module dimension>, <determinant formula for Specht module dimension>, and <Greene–Nijenhuis–Wilf hook walk>.
Back to article page
