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For any partition of an integer $\lambda ⊢n$, the number $f^{\lambda }$ of standard Young tableaux of shape $\lambda$ isThe 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 Etingof et al., Introduction to Representation Theory, §5.17, Theorem 5.17.1.

In the Young diagram of $(3,2)$, the hook length array $[[4,3,1],[2,1]]$ givesThis 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 $n\ge 1$, 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 §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: ^[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.

For a target corner $(\alpha ,\beta )$ in the Greene–Nijenhuis–Wilf hook walk, put $A_i=h(i,\beta )-1$ and $B_j=h(\alpha ,j)-1$. The probability of terminating there from $(a,b)$ factors as $F_a{G}_b$, where $F_{\alpha }=G_{\beta }=1$, $F_a=A_a^{-1}{\prod }_{i=a+1}^{\alpha -1}(1+A_i^{-1})$ for $a<\alpha$, and $G_b=B_b^{-1}{\prod }_{j=b+1}^{\beta -1}(1+B_j^{-1})$ for $b<\beta$. It is zero when the start is outside the target's northwest rectangle. The factorization uses $h(i,j)-1=A_i+B_j$ and the first-step recurrence.

For a cell $x=(i,j)$ in a Young diagram, its hook of a Young diagram consists of $x$, all cells to its right in row $i$, and all cells below it in column $j$. If ${\lambda }^{\prime }$ is the conjugate partition, its hook length isThe hook-length formula is

In the Greene–Nijenhuis–Wilf hook walk, a walk at a noncorner cell chooses uniformly one of the other $h(i,j)-1$ cells in its hook, and stops at a corner. Moves are strictly rightwards or downwards, so termination is certain. A terminal cell must be a Removable node of a Young diagram. For a chosen corner $(\alpha ,\beta )$, its probability is zero unless the start satisfies $a\le \alpha$ and $b\le \beta$.

For a start in that northwest rectangle, defineThe corner conditions are ${\lambda }_{\alpha }=\beta$ and ${\lambda }_{\beta }^{\prime }=\alpha$, so $A_{\alpha }=B_{\beta }=0$, and all earlier $A_i,B_j$ are positive. DefineandThen the closed conditional probability isIn particular, if both inequalities are strict, this isThe separate definitions of $F_{\alpha },G_{\beta }$ handle starts in the terminal row, in the terminal column, or at the corner itself without division by zero.

To prove the formula, the product definitions give the identitieswith the terminal zero identities interpreted as empty sums. For example, summing the $F$ recurrence backwards gives ${\sum }_{r=i+1}^{\alpha }{F}_r={\prod }_{r=i+1}^{\alpha -1}(1+A_r^{-1})$. AlsoThe first-step equation for the hook walk terminal probability isJumps beyond the target rectangle contribute zero. Substituting $p_{i,j}=F_i{G}_j$ proves this equation, and $p_{\alpha ,\beta }=1$ gives its boundary value. Since every step increases a coordinate, backward recursion determines the probability uniquely. This proves the claimed expression.