Lagmental Vicfred

The Hook-Length Formula Counts Standard Young Tableaux by Vicfred

Last updated: Tue 13 July 2010

tags: mathematics, integer partitions, Young tableaux, q-analogues

The number of standard tableaux of shape lambda is n factorial divided by the product of all hook lengths. A small computation will anchor the general statement before the abstraction takes over.

Notation

A partition \(\lambda\vdash n\) is both a decreasing sequence and a Ferrers diagram. Statistics such as hook lengths \(h_{ij}\) turn the diagram into exact product formulas.

$$ f^\lambda=\#\{\text{standard Young tableaux of shape }\lambda\} $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ f^\lambda=\frac{n!}{\prod_{(i,j)\in\lambda}h_{ij}} $$

Stress the formula

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \lambda=(3,2):\qquad(h_{ij})=\begin{array}{ccc}4&3&1\\2&1\end{array},\qquad f^{(3,2)}=\frac{5!}{4\cdot3\cdot1\cdot2\cdot1}=5 $$

Interpretation

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad f^\lambda=\#\{\text{standard Young tableaux of shape }\lambda\},\\[5pt] \mathsf{C}\;&:\quad f^\lambda=\frac{n!}{\prod_{(i,j)\in\lambda}h_{ij}}. \end{aligned} $$

Limit of the argument

Partitions forget order, compositions retain it, and tableaux add labels subject to row and column rules. Interchanging these objects changes the count.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f^\lambda=\frac{n!}{\prod_{(i,j)\in\lambda}h_{ij}} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 18 May 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.